START-INFO-DIR-ENTRY
* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
END-INFO-DIR-ENTRY
GNU MP
******
This manual describes how to install and use the GNU multiple precision
arithmetic library, version 4.1.
Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
2001, 2002 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.1 or any later version published by the Free Software Foundation;
with no Invariant Sections, with the Front-Cover Texts being "A GNU
Manual", and with the Back-Cover Texts being "You have freedom to copy
and modify this GNU Manual, like GNU software". A copy of the license
is included in *Note GNU Free Documentation License::.
...Table of Contents...
GNU MP Copying Conditions
*************************
This library is "free"; this means that everyone is free to use it
and free to redistribute it on a free basis. The library is not in the
public domain; it is copyrighted and there are restrictions on its
distribution, but these restrictions are designed to permit everything
that a good cooperating citizen would want to do. What is not allowed
is to try to prevent others from further sharing any version of this
library that they might get from you.
Specifically, we want to make sure that you have the right to give
away copies of the library, that you receive source code or else can
get it if you want it, that you can change this library or use pieces
of it in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to
deprive anyone else of these rights. For example, if you distribute
copies of the GNU MP library, you must give the recipients all the
rights that you have. You must make sure that they, too, receive or
can get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone
finds out that there is no warranty for the GNU MP library. If it is
modified by someone else and passed on, we want their recipients to
know that what they have is not what we distributed, so that any
problems introduced by others will not reflect on our reputation.
The precise conditions of the license for the GNU MP library are
found in the Lesser General Public License version 2.1 that accompanies
the source code, see `COPYING.LIB'. Certain demonstration programs are
provided under the terms of the plain General Public License version 2,
see `COPYING'.
Introduction to GNU MP
**********************
GNU MP is a portable library written in C for arbitrary precision
arithmetic on integers, rational numbers, and floating-point numbers.
It aims to provide the fastest possible arithmetic for all applications
that need higher precision than is directly supported by the basic C
types.
Many applications use just a few hundred bits of precision; but some
applications may need thousands or even millions of bits. GMP is
designed to give good performance for both, by choosing algorithms
based on the sizes of the operands, and by carefully keeping the
overhead at a minimum.
The speed of GMP is achieved by using fullwords as the basic
arithmetic type, by using sophisticated algorithms, by including
carefully optimized assembly code for the most common inner loops for
many different CPUs, and by a general emphasis on speed (as opposed to
simplicity or elegance).
There is carefully optimized assembly code for these CPUs: ARM, DEC
Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2 and Athlon,
Hitachi SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium
Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola
MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64,
National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC,
generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some
optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and
Pyramid AP/XP.
There is a mailing list for GMP users. To join it, send a mail to
with the word `subscribe' in the message *body*
(not in the subject line).
For up-to-date information on GMP, please see the GMP web pages at
`http://swox.com/gmp/'
The latest version of the library is available at
`ftp://ftp.gnu.org/gnu/gmp'
Many sites around the world mirror `ftp.gnu.org', please use a mirror
near you, see `http://www.gnu.org/order/ftp.html' for a full list.
How to use this Manual
======================
Everyone should read *Note GMP Basics::. If you need to install the
library yourself, then read *Note Installing GMP::. If you have a
system with multiple ABIs, then read *Note ABI and ISA::, for the
compiler options that must be used on applications.
The rest of the manual can be used for later reference, although it
is probably a good idea to glance through it.
Installing GMP
**************
GMP has an autoconf/automake/libtool based configuration system. On
a Unix-like system a basic build can be done with
./configure
make
Some self-tests can be run with
make check
And you can install (under `/usr/local' by default) with
make install
If you experience problems, please report them to .
See *Note Reporting Bugs::, for information on what to include in
useful bug reports.
Build Options
=============
All the usual autoconf configure options are available, run
`./configure --help' for a summary. The file `INSTALL.autoconf' has
some generic installation information too.
Non-Unix Systems
`configure' requires various Unix-like tools. On an MS-DOS system
DJGPP can be used, and on MS Windows Cygwin or MINGW can be used,
`http://www.cygnus.com/cygwin'
`http://www.delorie.com/djgpp'
`http://www.mingw.org'
The `macos' directory contains an unsupported port to MacOS 9 on
Power Macintosh, see `macos/README'. Note that MacOS X "Darwin"
should use the normal Unix-style `./configure'.
It might be possible to build without the help of `configure',
certainly all the code is there, but unfortunately you'll be on
your own.
Build Directory
To compile in a separate build directory, `cd' to that directory,
and prefix the configure command with the path to the GMP source
directory. For example
cd /my/build/dir
/my/sources/gmp-4.1/configure
Not all `make' programs have the necessary features (`VPATH') to
support this. In particular, SunOS and Slowaris `make' have bugs
that make them unable to build in a separate directory. Use GNU
`make' instead.
`--disable-shared', `--disable-static'
By default both shared and static libraries are built (where
possible), but one or other can be disabled. Shared libraries
result in smaller executables and permit code sharing between
separate running processes, but on some CPUs are slightly slower,
having a small cost on each function call.
Native Compilation, `--build=CPU-VENDOR-OS'
For normal native compilation, the system can be specified with
`--build'. By default `./configure' uses the output from running
`./config.guess'. On some systems `./config.guess' can determine
the exact CPU type, on others it will be necessary to give it
explicitly. For example,
./configure --build=ultrasparc-sun-solaris2.7
In all cases the `OS' part is important, since it controls how
libtool generates shared libraries. Running `./config.guess' is
the simplest way to see what it should be, if you don't know
already.
Cross Compilation, `--host=CPU-VENDOR-OS'
When cross-compiling, the system used for compiling is given by
`--build' and the system where the library will run is given by
`--host'. For example when using a FreeBSD Athlon system to build
GNU/Linux m68k binaries,
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
Compiler tools are sought first with the host system type as a
prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
plain `ranlib'. This makes it possible for a set of
cross-compiling tools to co-exist with native tools. The prefix
is the argument to `--host', and this can be an alias, such as
`m68k-linux'. But note that tools don't have to be setup this
way, it's enough to just have a `PATH' with a suitable
cross-compiling `cc' etc.
Compiling for a different CPU in the same family as the build
system is a form of cross-compilation, though very possibly this
would merely be special options on a native compiler. In any case
`./configure' avoids depending on being able to run code on the
build system, which is important when creating binaries for a
newer CPU since they very possibly won't run on the build system.
In all cases the compiler must be able to produce an executable
(of whatever format) from a standard C `main'. Although only
object files will go to make up `libgmp', `./configure' uses
linking tests for various purposes, such as determining what
functions are available on the host system.
Currently a warning is given unless an explicit `--build' is used
when cross-compiling, because it may not be possible to correctly
guess the build system type if the `PATH' has only a
cross-compiling `cc'.
Note that the `--target' option is not appropriate for GMP. It's
for use when building compiler tools, with `--host' being where
they will run, and `--target' what they'll produce code for.
Ordinary programs or libraries like GMP are only interested in the
`--host' part, being where they'll run. (Some past versions of
GMP used `--target' incorrectly.)
CPU types
In general, if you want a library that runs as fast as possible,
you should configure GMP for the exact CPU type your system uses.
However, this may mean the binaries won't run on older members of
the family, and might run slower on other members, older or newer.
The best idea is always to build GMP for the exact machine type
you intend to run it on.
The following CPUs have specific support. See `configure.in' for
details of what code and compiler options they select.
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
alphaev6, alphaev67, alphaev68
* Cray: c90, j90, t90, sv1
* HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w
* MIPS: mips, mips3, mips64
* Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
m68060, m68302, m68360, m88k, m88110
* POWER: power, power1, power2, power2sc
* PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
powerpc821, powerpc823, powerpc860,
* SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
* 80x86 family: i386, i486, i586, pentium, pentiummmx,
pentiumpro, pentium2, pentium3, pentium4, k6, k62, k63, athlon
* Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
z8k
CPUs not listed will use generic C code.
Generic C Build
If some of the assembly code causes problems, or if otherwise
desired, the generic C code can be selected with CPU `none'. For
example,
./configure --host=none-unknown-freebsd3.5
Note that this will run quite slowly, but it should be portable
and should at least make it possible to get something running if
all else fails.
`ABI'
On some systems GMP supports multiple ABIs (application binary
interfaces), meaning data type sizes and calling conventions. By
default GMP chooses the best ABI available, but a particular ABI
can be selected. For example
./configure --host=mips64-sgi-irix6 ABI=n32
See *Note ABI and ISA::, for the available choices on relevant
CPUs, and what applications need to do.
`CC', `CFLAGS'
By default the C compiler used is chosen from among some likely
candidates, with `gcc' normally preferred if it's present. The
usual `CC=whatever' can be passed to `./configure' to choose
something different.
For some systems, default compiler flags are set based on the CPU
and compiler. The usual `CFLAGS="-whatever"' can be passed to
`./configure' to use something different or to set good flags for
systems GMP doesn't otherwise know.
The `CC' and `CFLAGS' used are printed during `./configure', and
can be found in each generated `Makefile'. This is the easiest way
to check the defaults when considering changing or adding
something.
Note that when `CC' and `CFLAGS' are specified on a system
supporting multiple ABIs it's important to give an explicit
`ABI=whatever', since GMP can't determine the ABI just from the
flags and won't be able to select the correct assembler code.
If just `CC' is selected then normal default `CFLAGS' for that
compiler will be used (if GMP recognises it). For example
`CC=gcc' can be used to force the use of GCC, with default flags
(and default ABI).
`CPPFLAGS'
Any flags like `-D' defines or `-I' includes required by the
preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
Compiling is done with both `CPPFLAGS' and `CFLAGS', but
preprocessing uses just `CPPFLAGS'. This distinction is because
most preprocessors won't accept all the flags the compiler does.
Preprocessing is done separately in some configure tests, and in
the `ansi2knr' support for K&R compilers.
C++ Support, `--enable-cxx'
C++ support in GMP can be enabled with `--enable-cxx', in which
case a C++ compiler will be required. As a convenience
`--enable-cxx=detect' can be used to enable C++ support only if a
compiler can be found. The C++ support consists of a library
`libgmpxx.la' and header file `gmpxx.h'.
A separate `libgmpxx.la' has been adopted rather than having C++
objects within `libgmp.la' in order to ensure dynamic linked C
programs aren't bloated by a dependency on the C++ standard
library, and to avoid any chance that the C++ compiler could be
required when linking plain C programs.
`libgmpxx.la' will use certain internals from `libgmp.la' and can
only be expected to work with `libgmp.la' from the same GMP
version. Future changes to the relevant internals will be
accompanied by renaming, so a mismatch will cause unresolved
symbols rather than perhaps mysterious misbehaviour.
In general `libgmpxx.la' will be usable only with the C++ compiler
that built it, since name mangling and runtime support are usually
incompatible between different compilers.
`CXX', `CXXFLAGS'
When C++ support is enabled, the C++ compiler and its flags can be
set with variables `CXX' and `CXXFLAGS' in the usual way. The
default for `CXX' is the first compiler that works from a list of
likely candidates, with `g++' normally preferred when available.
The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
`-g', then for `g++' either `-g -O2' or `-O2', or for other
compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
when using `gcc' and `g++' together, since the flags for `gcc' will
usually suit `g++'.
It's important that the C and C++ compilers match, meaning their
startup and runtime support routines are compatible and that they
generate code in the same ABI (if there's a choice of ABIs on the
system). `./configure' isn't currently able to check these things
very well itself, so for that reason `--disable-cxx' is the
default, to avoid a build failure due to a compiler mismatch.
Perhaps this will change in the future.
Incidentally, it's normally not good enough to set `CXX' to the
same as `CC'. Although `gcc' for instance recognises `foo.cc' as
C++ code, only `g++' will invoke the linker the right way when
building an executable or shared library from object files.
Temporary Memory, `--enable-alloca='
GMP allocates temporary workspace using one of the following three
methods, which can be selected with for instance
`--enable-alloca=malloc-reentrant'.
* `alloca' - C library or compiler builtin.
* `malloc-reentrant' - the heap, in a re-entrant fashion.
* `malloc-notreentrant' - the heap, with global variables.
For convenience, the following choices are also available.
`--disable-alloca' is the same as `--enable-alloca=no'.
* `yes' - a synonym for `alloca'.
* `no' - a synonym for `malloc-reentrant'.
* `reentrant' - `alloca' if available, otherwise
`malloc-reentrant'. This is the default.
* `notreentrant' - `alloca' if available, otherwise
`malloc-notreentrant'.
`alloca' is reentrant and fast, and is recommended, but when
working with large numbers it can overflow the available stack
space, in which case one of the two malloc methods will need to be
used. Alternately it might be possible to increase available
stack with `limit', `ulimit' or `setrlimit', or under DJGPP with
`stubedit' or `_stklen'. Note that depending on the system the
only indication of stack overflow might be a segmentation
violation.
`malloc-reentrant' is, as the name suggests, reentrant and thread
safe, but `malloc-notreentrant' is faster and should be used if
reentrancy is not required.
The two malloc methods in fact use the memory allocation functions
selected by `mp_set_memory_functions', these being `malloc' and
friends by default. *Note Custom Allocation::.
An additional choice `--enable-alloca=debug' is available, to help
when debugging memory related problems (*note Debugging::).
FFT Multiplication, `--disable-fft'
By default multiplications are done using Karatsuba, 3-way
Toom-Cook, and Fermat FFT. The FFT is only used on large to very
large operands and can be disabled to save code size if desired.
Berkeley MP, `--enable-mpbsd'
The Berkeley MP compatibility library (`libmp') and header file
(`mp.h') are built and installed only if `--enable-mpbsd' is used.
*Note BSD Compatible Functions::.
MPFR, `--enable-mpfr'
The optional MPFR functions are built and installed only if
`--enable-mpfr' is used. These are in a separate library
`libmpfr.a' and are documented separately too (*note Introduction
to MPFR: (mpfr)Introduction to MPFR.).
Assertion Checking, `--enable-assert'
This option enables some consistency checking within the library.
This can be of use while debugging, *note Debugging::.
Execution Profiling, `--enable-profiling=prof/gprof'
Profiling support can be enabled either for `prof' or `gprof'.
This adds `-p' or `-pg' respectively to `CFLAGS', and for some
systems adds corresponding `mcount' calls to the assembler code.
*Note Profiling::.
`MPN_PATH'
Various assembler versions of each mpn subroutines are provided.
For a given CPU, a search is made though a path to choose a
version of each. For example `sparcv8' has
MPN_PATH="sparc32/v8 sparc32 generic"
which means look first for v8 code, then plain sparc32 (which is
v7), and finally fall back on generic C. Knowledgeable users with
special requirements can specify a different path. Normally this
is completely unnecessary.
Demonstration Programs
The `demos' subdirectory has some sample programs using GMP. These
aren't built or installed, but there's a `Makefile' with rules for
them. For instance,
make pexpr
./pexpr 68^975+10
Documentation
The document you're now reading is `gmp.texi'. The usual automake
targets are available to make PostScript `gmp.ps' and/or DVI
`gmp.dvi'.
HTML can be produced with `makeinfo --html', see *Note Generating
HTML: (texinfo)makeinfo html. Or alternately `texi2html', see
*Note Texinfo to HTML: (texi2html)Top.
PDF can be produced with `texi2dvi --pdf' (*note PDF: (texinfo)PDF
Output.) or with `pdftex'.
Some supplementary notes can be found in the `doc' subdirectory.
ABI and ISA
===========
ABI (Application Binary Interface) refers to the calling conventions
between functions, meaning what registers are used and what sizes the
various C data types are. ISA (Instruction Set Architecture) refers to
the instructions and registers a CPU has available.
Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
defined, the latter for compatibility with older CPUs in the family.
GMP supports some CPUs like this in both ABIs. In fact within GMP
`ABI' means a combination of chip ABI, plus how GMP chooses to use it.
For example in some 32-bit ABIs, GMP may support a limb as either a
32-bit `long' or a 64-bit `long long'.
By default GMP chooses the best ABI available for a given system,
and this generally gives significantly greater speed. But an ABI can
be chosen explicitly to make GMP compatible with other libraries, or
particular application requirements. For example,
./configure ABI=32
In all cases it's vital that all object code used in a given program
is compiled for the same ABI.
Usually a limb is implemented as a `long'. When a `long long' limb
is used this is encoded in the generated `gmp.h'. This is convenient
for applications, but it does mean that `gmp.h' will vary, and can't be
just copied around. `gmp.h' remains compiler independent though, since
all compilers for a particular ABI will be expected to use the same
limb type.
Currently no attempt is made to follow whatever conventions a system
has for installing library or header files built for a particular ABI.
This will probably only matter when installing multiple builds of GMP,
and it might be as simple as configuring with a special `libdir', or it
might require more than that. Note that builds for different ABIs need
to done separately, with a fresh `./configure' and `make' each.
HPPA 2.0 (`hppa2.0*')
`ABI=2.0w'
The 2.0w ABI uses 64-bit limbs and pointers and is available
on HP-UX 11 or up when using `cc'. `gcc' support for this is
in progress. Applications must be compiled with
cc +DD64
`ABI=2.0n'
The 2.0n ABI means the 32-bit HPPA 1.0 ABI but with a 64-bit
limb using `long long'. This is available on HP-UX 10 or up
when using `cc'. No `gcc' support is planned for this.
Applications must be compiled with
cc +DA2.0 +e
`ABI=1.0'
HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit
HPPA 1.0 ABI. No special compiler options are needed for
applications.
All three ABIs are available for CPUs `hppa2.0w' and `hppa2.0', but
for CPU `hppa2.0n' only 2.0n or 1.0 are allowed.
MIPS under IRIX 6 (`mips*-*-irix[6789]')
IRIX 6 supports the n32 and 64 ABIs and always has a 64-bit MIPS 3
or better CPU. In both these ABIs GMP uses a 64-bit limb. A new
enough `gcc' is required (2.95 for instance).
`ABI=n32'
The n32 ABI is 32-bit pointers and integers, but with a
64-bit limb using a `long long'. Applications must be
compiled with
gcc -mabi=n32
cc -n32
`ABI=64'
The 64-bit ABI is 64-bit pointers and integers. Applications
must be compiled with
gcc -mabi=64
cc -64
Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have
the necessary support for n32 or 64 and so only gets a 32-bit limb
and the MIPS 2 code.
PowerPC 64 (`powerpc64*')
`ABI=aix64'
The AIX 64 ABI uses 64-bit limbs and pointers and is
available on systems `powerpc64*-*-aix*'. Applications must
be compiled (and linked) with
gcc -maix64
xlc -q64
`ABI=32L'
This uses the 32-bit ABI but a 64-bit limb using GCC `long
long' in 64-bit registers. Applications must be compiled with
gcc -mpowerpc64
`ABI=32'
This is the basic 32-bit PowerPC ABI. No special compiler
options are needed for applications.
Sparc V9 (`sparcv9' and `ultrasparc*')
`ABI=64'
The 64-bit V9 ABI is available on Solaris 2.7 and up and
GNU/Linux. GCC 2.95 or up, or Sun `cc' is required.
Applications must be compiled with
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc -xarch=v9
`ABI=32'
On Solaris 2.6 and earlier, and on Solaris 2.7 with the
kernel in 32-bit mode, only the plain V8 32-bit ABI can be
used, since the kernel doesn't save all registers. GMP still
uses as much of the V9 ISA as it can in these circumstances.
No special compiler options are required for applications,
though using something like the following requesting V9 code
within the V8 ABI is recommended.
gcc -mv8plus
cc -xarch=v8plus
`gcc' 2.8 and earlier only supports `-mv8' though.
Don't be confused by the names of these sparc `-m' and `-x'
options, they're called `arch' but they effectively control the
ABI.
On Solaris 2.7 with the kernel in 32-bit-mode, a normal native
build will reject `ABI=64' because the resulting executables won't
run. `ABI=64' can still be built if desired by making it look
like a cross-compile, for example
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
Notes for Package Builds
========================
GMP should present no great difficulties for packaging in a binary
distribution.
Libtool is used to build the library and `-version-info' is set
appropriately, having started from `3:0:0' in GMP 3.0. The GMP 4 series
will be upwardly binary compatible in each release and will be upwardly
binary compatible with all of the GMP 3 series. Additional function
interfaces may be added in each release, so on systems where libtool
versioning is not fully checked by the loader an auxiliary mechanism
may be needed to express that a dynamic linked application depends on a
new enough GMP.
An auxiliary mechanism may also be needed to express that
`libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires
`libgmp.la' from the same GMP version, since this is not done by the
libtool versioning, nor otherwise. A mismatch will result in
unresolved symbols from the linker, or perhaps the loader.
Using `DESTDIR' or a `prefix' override with `make install' and a
shared `libgmpxx' may run into a libtool relinking problem, see *Note
Known Build Problems::.
When building a package for a CPU family, care should be taken to use
`--host' (or `--build') to choose the least common denominator among
the CPUs which might use the package. For example this might
necessitate `i386' for x86s, or plain `sparc' (meaning V7) for SPARCs.
Users who care about speed will want GMP built for their exact CPU
type, to make use of the available optimizations. Providing a way to
suitably rebuild a package may be useful. This could be as simple as
making it possible for a user to omit `--build' (and `--host') so
`./config.guess' will detect the CPU. But a way to manually specify a
`--build' will be wanted for systems where `./config.guess' is inexact.
Note that `gmp.h' is a generated file, and will be architecture and
ABI dependent.
Notes for Particular Systems
============================
AIX 3 and 4
On systems `*-*-aix[34]*' shared libraries are disabled by
default, since some versions of the native `ar' fail on the
convenience libraries used. A shared build can be attempted with
./configure --enable-shared --disable-static
Note that the `--disable-static' is necessary because in a shared
build libtool makes `libgmp.a' a symlink to `libgmp.so',
apparently for the benefit of old versions of `ld' which only
recognise `.a', but unfortunately this is done even if a fully
functional `ld' is available.
ARM
On systems `arm*-*-*', versions of GCC up to and including 2.95.3
have a bug in unsigned division, giving wrong results for some
operands. GMP `./configure' will demand GCC 2.95.4 or later.
Microsoft Windows
On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
GMP builds only a static library, but a DLL can be built instead
using
./configure --disable-static --enable-shared
Static and DLL libraries can't both be built, since certain export
directives in `gmp.h' must be different. `--enable-cxx' cannot be
used when building a DLL, since libtool doesn't currently support
C++ DLLs. This might change in the future.
GCC is recommended for compiling GMP, but the resulting DLL can be
used with any compiler. On mingw only the standard Windows
libraries will be needed, on Cygwin the usual cygwin runtime will
be required.
Motorola 68k CPU Types
`m68k' is taken to mean 68000. `m68020' or higher will give a
performance boost on applicable CPUs. `m68360' can be used for
CPU32 series chips. `m68302' can be used for "Dragonball" series
chips, though this is merely a synonym for `m68000'.
OpenBSD 2.6
`m4' in this release of OpenBSD has a bug in `eval' that makes it
unsuitable for `.asm' file processing. `./configure' will detect
the problem and either abort or choose another m4 in the `PATH'.
The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
Power CPU Types
In GMP, CPU types `power*' and `powerpc*' will each use
instructions not available on the other, so it's important to
choose the right one for the CPU that will be used. Currently GMP
has no assembler code support for using just the common
instruction subset. To get executables that run on both, the
current suggestion is to use the generic C code (CPU `none'),
possibly with appropriate compiler options (like `-mcpu=common' for
`gcc'). CPU `rs6000' (which is not a CPU but a family of
workstations) is accepted by `config.sub', but is currently
equivalent to `none'.
Sparc CPU Types
`sparcv8' or `supersparc' on relevant systems will give a
significant performance increase over the V7 code.
SunOS 4
`/usr/bin/m4' lacks various features needed to process `.asm'
files, and instead `./configure' will automatically use
`/usr/5bin/m4', which we believe is always available (if not then
use GNU m4).
x86 CPU Types
`i386' selects generic code which will run reasonably well on all
x86 chips.
`i586', `pentium' or `pentiummmx' code is good for the intended P5
Pentium chips, but quite slow when run on Intel P6 class chips
(PPro, P-II, P-III). `i386' is a better choice when making
binaries that must run on both.
`pentium4' and an SSE2 capable assembler are important for best
results on Pentium 4. The specific code is for instance roughly a
2x to 3x speedup over the generic `i386' code.
x86 MMX and SSE2 Code
If the CPU selected has MMX code but the assembler doesn't support
it, a warning is given and non-MMX code is used instead. This
will be an inferior build, since the MMX code that's present is
there because it's faster than the corresponding plain integer
code. The same applies to SSE2.
Old versions of `gas' don't support MMX instructions, in particular
version 1.92.3 that comes with FreeBSD 2.2.8 doesn't (and
unfortunately there's no newer assembler for that system).
Solaris 2.6 and 2.7 `as' generate incorrect object code for
register to register `movq' instructions, and so can't be used for
MMX code. Install a recent `gas' if MMX code is wanted on these
systems.
x86 GCC `-march=pentiumpro'
GCC 2.95.2 and 2.95.3 miscompiled some versions of `mpz/powm.c'
when `-march=pentiumpro' was used, so for relevant CPUs that
option is only in the default `CFLAGS' for GCC 2.95.4 and up.
Known Build Problems
====================
You might find more up-to-date information at `http://swox.com/gmp/'.
DJGPP
The DJGPP port of `bash' 2.03 is unable to run the `configure'
script, it exits silently, having died writing a preamble to
`config.log'. Use `bash' 2.04 or higher.
`make all' was found to run out of memory during the final
`libgmp.la' link on one system tested, despite having 64Mb
available. A separate `make libgmp.la' helped, perhaps recursing
into the various subdirectories uses up memory.
`DESTDIR' and shared `libgmpxx'
`make install DESTDIR=/my/staging/area' or the same with a `prefix'
override to install to a temporary directory is not fully
supported by current versions of libtool when building a shared
version of a library which depends on another being built at the
same time, like `libgmpxx' and `libgmp'.
The problem is that `libgmpxx' is relinked at the install stage to
ensure that if the system puts a hard-coded path to `libgmp' within
`libgmpxx' then that path will be correct. Naturally the linker is
directed to look only at the final location, not the staging area,
so if `libgmp' is not already in that final location then the link
will fail.
On systems which don't hard-code library paths, for instance SVR4
style systems such as GNU/Linux, a workaround is to insert a
suitable `-L' in the `relink_command' of `libgmpxx.la' after
building but before installing. This can be automated with
something like
sed '/^relink_command/s:libgmp.la:-L /my/staging/area libgmp.la:' \
libgmpxx.new
mv libgmpxx.new libgmpxx.la
GNU binutils `strip'
GNU binutils `strip' should not be used on the static libraries
`libgmp.a' and `libmp.a', neither directly nor via `make
install-strip'. It can be used on the shared libraries
`libgmp.so' and `libmp.so' though.
Currently (binutils 2.10.0), `strip' unpacks an archive then
operates on the files, but GMP contains multiple object files of
the same name (eg. three versions of `init.o'), and they overwrite
each other, leaving only the one that happens to be last.
If stripped static libraries are wanted, the suggested workaround
is to build normally, strip the separate object files, and do
another `make all' to rebuild. Alternately `CFLAGS' with `-g'
omitted can always be used if it's just debugging which is
unwanted.
`make' syntax error
On certain versions of SCO OpenServer 5 and IRIX 6.5 the native
`make' is unable to handle the long dependencies list for
`libgmp.la'. The symptom is a "syntax error" on the following
line of the top-level `Makefile'.
libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
Either use GNU Make, or as a workaround remove
`$(libgmp_la_DEPENDENCIES)' from that line (which will make the
initial build work, but if any recompiling is done `libgmp.la'
might not be rebuilt).
NeXT prior to 3.3
The system compiler on old versions of NeXT was a massacred and
old GCC, even if it called itself `cc'. This compiler cannot be
used to build GMP, you need to get a real GCC, and install that.
(NeXT may have fixed this in release 3.3 of their system.)
POWER and PowerPC
Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP
on POWER or PowerPC. If you want to use GCC for these machines,
get GCC 2.7.2.1 (or later).
Sequent Symmetry
Use the GNU assembler instead of the system assembler, since the
latter has serious bugs.
Solaris 2.6
The system `sed' prints an error "Output line too long" when
libtool builds `libgmp.la'. This doesn't seem to cause any
obvious ill effects, but GNU `sed' is recommended, to avoid any
doubt.
Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32
A shared library build of GMP seems to fail in this combination,
it builds but then fails the tests, apparently due to some
incorrect data relocations within `gmp_randinit_lc_2exp_size'.
The exact cause is unknown, `--disable-shared' is recommended.
Windows DLL test programs
When creating a DLL version of `libgmp', libtool creates wrapper
scripts like `t-mul' for programs that would normally be
`t-mul.exe', in order to setup the right library paths etc. This
works fine, but the absence of `t-mul.exe' etc causes `make' to
think they need recompiling every time, which is an annoyance when
re-running a `make check'.
GMP Basics
**********
*Using functions, macros, data types, etc. not documented in this
manual is strongly discouraged. If you do so your application is
guaranteed to be incompatible with future versions of GMP.*
Headers and Libraries
=====================
All declarations needed to use GMP are collected in the include file
`gmp.h'. It is designed to work with both C and C++ compilers.
#include
Note however that prototypes for GMP functions with `FILE *'
parameters are only provided if `' is included too.
#include
#include
Likewise `' (or `') is required for prototypes
with `va_list' parameters, such as `gmp_vprintf'. And `'
for prototypes with `struct obstack' parameters, such as
`gmp_obstack_printf', when available.
All programs using GMP must link against the `libgmp' library. On a
typical Unix-like system this can be done with `-lgmp', for example
gcc myprogram.c -lgmp
GMP C++ functions are in a separate `libgmpxx' library. This is
built and installed if C++ support has been enabled (*note Build
Options::). For example,
g++ mycxxprog.cc -lgmpxx -lgmp
GMP is built using Libtool and an application can use that to link
if desired, *note Shared library support for GNU: (libtool)Top.
If GMP has been installed to a non-standard location then it may be
necessary to use `-I' and `-L' compiler options to point to the right
directories, and some sort of run-time path for a shared library.
Consult your compiler documentation, for instance *Note Introduction:
(gcc)Top.
Nomenclature and Types
======================
In this manual, "integer" usually means a multiple precision integer, as
defined by the GMP library. The C data type for such integers is
`mpz_t'. Here are some examples of how to declare such integers:
mpz_t sum;
struct foo { mpz_t x, y; };
mpz_t vec[20];
"Rational number" means a multiple precision fraction. The C data type
for these fractions is `mpq_t'. For example:
mpq_t quotient;
"Floating point number" or "Float" for short, is an arbitrary precision
mantissa with a limited precision exponent. The C data type for such
objects is `mpf_t'.
A "limb" means the part of a multi-precision number that fits in a
single machine word. (We chose this word because a limb of the human
body is analogous to a digit, only larger, and containing several
digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
is `mp_limb_t'.
Function Classes
================
There are six classes of functions in the GMP library:
1. Functions for signed integer arithmetic, with names beginning with
`mpz_'. The associated type is `mpz_t'. There are about 150
functions in this class.
2. Functions for rational number arithmetic, with names beginning with
`mpq_'. The associated type is `mpq_t'. There are about 40
functions in this class, but the integer functions can be used for
arithmetic on the numerator and denominator separately.
3. Functions for floating-point arithmetic, with names beginning with
`mpf_'. The associated type is `mpf_t'. There are about 60
functions is this class.
4. Functions compatible with Berkeley MP, such as `itom', `madd', and
`mult'. The associated type is `MINT'.
5. Fast low-level functions that operate on natural numbers. These
are used by the functions in the preceding groups, and you can
also call them directly from very time-critical user programs.
These functions' names begin with `mpn_'. The associated type is
array of `mp_limb_t'. There are about 30 (hard-to-use) functions
in this class.
6. Miscellaneous functions. Functions for setting up custom
allocation and functions for generating random numbers.
Variable Conventions
====================
GMP functions generally have output arguments before input
arguments. This notation is by analogy with the assignment operator.
The BSD MP compatibility functions are exceptions, having the output
arguments last.
GMP lets you use the same variable for both input and output in one
call. For example, the main function for integer multiplication,
`mpz_mul', can be used to square `x' and put the result back in `x' with
mpz_mul (x, x, x);
Before you can assign to a GMP variable, you need to initialize it
by calling one of the special initialization functions. When you're
done with a variable, you need to clear it out, using one of the
functions for that purpose. Which function to use depends on the type
of variable. See the chapters on integer functions, rational number
functions, and floating-point functions for details.
A variable should only be initialized once, or at least cleared
between each initialization. After a variable has been initialized, it
may be assigned to any number of times.
For efficiency reasons, avoid excessive initializing and clearing.
In general, initialize near the start of a function and clear near the
end. For example,
void
foo (void)
{
mpz_t n;
int i;
mpz_init (n);
for (i = 1; i < 100; i++)
{
mpz_mul (n, ...);
mpz_fdiv_q (n, ...);
...
}
mpz_clear (n);
}
Parameter Conventions
=====================
When a GMP variable is used as a function parameter, it's
effectively a call-by-reference, meaning if the function stores a value
there it will change the original in the caller. Parameters which are
input-only can be designated `const' to provoke a compiler error or
warning on attempting to modify them.
When a function is going to return a GMP result, it should designate
a parameter that it sets, like the library functions do. More than one
value can be returned by having more than one output parameter, again
like the library functions. A `return' of an `mpz_t' etc doesn't
return the object, only a pointer, and this is almost certainly not
what's wanted.
Here's an example accepting an `mpz_t' parameter, doing a
calculation, and storing the result to the indicated parameter.
void
foo (mpz_t result, const mpz_t param, unsigned long n)
{
unsigned long i;
mpz_mul_ui (result, param, n);
for (i = 1; i < n; i++)
mpz_add_ui (result, result, i*7);
}
int
main (void)
{
mpz_t r, n;
mpz_init (r);
mpz_init_set_str (n, "123456", 0);
foo (r, n, 20L);
gmp_printf ("%Zd\n", r);
return 0;
}
`foo' works even if the mainline passes the same variable for
`param' and `result', just like the library functions. But sometimes
it's tricky to make that work, and an application might not want to
bother supporting that sort of thing.
For interest, the GMP types `mpz_t' etc are implemented as
one-element arrays of certain structures. This is why declaring a
variable creates an object with the fields GMP needs, but then using it
as a parameter passes a pointer to the object. Note that the actual
fields in each `mpz_t' etc are for internal use only and should not be
accessed directly by code that expects to be compatible with future GMP
releases.
Memory Management
=================
The GMP types like `mpz_t' are small, containing only a couple of
sizes, and pointers to allocated data. Once a variable is initialized,
GMP takes care of all space allocation. Additional space is allocated
whenever a variable doesn't have enough.
`mpz_t' and `mpq_t' variables never reduce their allocated space.
Normally this is the best policy, since it avoids frequent reallocation.
Applications that need to return memory to the heap at some particular
point can use `mpz_realloc2', or clear variables no longer needed.
`mpf_t' variables, in the current implementation, use a fixed amount
of space, determined by the chosen precision and allocated at
initialization, so their size doesn't change.
All memory is allocated using `malloc' and friends by default, but
this can be changed, see *Note Custom Allocation::. Temporary memory
on the stack is also used (via `alloca'), but this can be changed at
build-time if desired, see *Note Build Options::.
Reentrancy
==========
GMP is reentrant and thread-safe, with some exceptions:
* If configured with `--enable-alloca=malloc-notreentrant' (or with
`--enable-alloca=notreentrant' when `alloca' is not available),
then naturally GMP is not reentrant.
* `mpf_set_default_prec' and `mpf_init' use a global variable for the
selected precision. `mpf_init2' can be used instead.
* `mpz_random' and the other old random number functions use a global
random state and are hence not reentrant. The newer random number
functions that accept a `gmp_randstate_t' parameter can be used
instead.
* `mp_set_memory_functions' uses global variables to store the
selected memory allocation functions.
* If the memory allocation functions set by a call to
`mp_set_memory_functions' (or `malloc' and friends by default) are
not reentrant, then GMP will not be reentrant either.
* If the standard I/O functions such as `fwrite' are not reentrant
then the GMP I/O functions using them will not be reentrant either.
* It's safe for two threads to read from the same GMP variable
simultaneously, but it's not safe for one to read while the
another might be writing, nor for two threads to write
simultaneously. It's not safe for two threads to generate a
random number from the same `gmp_randstate_t' simultaneously,
since this involves an update of that variable.
* On SCO systems the default `' macros use per-file static
variables and may not be reentrant, depending whether the compiler
optimizes away fetches from them. The GMP text-based input
functions are affected.
Useful Macros and Constants
===========================
- Global Constant: const int mp_bits_per_limb
The number of bits per limb.
- Macro: __GNU_MP_VERSION
- Macro: __GNU_MP_VERSION_MINOR
- Macro: __GNU_MP_VERSION_PATCHLEVEL
The major and minor GMP version, and patch level, respectively, as
integers. For GMP i.j, these numbers will be i, j, and 0,
respectively. For GMP i.j.k, these numbers will be i, j, and k,
respectively.
- Global Constant: const char * const gmp_version
The GMP version number, as a null-terminated string, in the form
"i.j" or "i.j.k". This release is "4.1".
Compatibility with older versions
=================================
This version of GMP is upwardly binary compatible with all 4.x and
3.x versions, and upwardly compatible at the source level with all 2.x
versions, with the following exceptions.
* `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
consistency with other `mpn' functions.
* `mpf_get_prec' counted precision slightly differently in GMP 3.0
and 3.0.1, but in 3.1 reverted to the 2.x style.
There are a number of compatibility issues between GMP 1 and GMP 2
that of course also apply when porting applications from GMP 1 to GMP
4. Please see the GMP 2 manual for details.
The Berkeley MP compatibility library (*note BSD Compatible
Functions::) is source and binary compatible with the standard `libmp'.
Efficiency
==========
Small operands
On small operands, the time for function call overheads and memory
allocation can be significant in comparison to actual calculation.
This is unavoidable in a general purpose variable precision
library, although GMP attempts to be as efficient as it can on
both large and small operands.
Static Linking
On some CPUs, in particular the x86s, the static `libgmp.a' should
be used for maximum speed, since the PIC code in the shared
`libgmp.so' will have a small overhead on each function call and
global data address. For many programs this will be
insignificant, but for long calculations there's a gain to be had.
Initializing and clearing
Avoid excessive initializing and clearing of variables, since this
can be quite time consuming, especially in comparison to otherwise
fast operations like addition.
A language interpreter might want to keep a free list or stack of
initialized variables ready for use. It should be possible to
integrate something like that with a garbage collector too.
Reallocations
An `mpz_t' or `mpq_t' variable used to hold successively increasing
values will have its memory repeatedly `realloc'ed, which could be
quite slow or could fragment memory, depending on the C library.
If an application can estimate the final size then `mpz_init2' or
`mpz_realloc2' can be called to allocate the necessary space from
the beginning (*note Initializing Integers::).
It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
is too small, since all functions will do a further reallocation
if necessary. Badly overestimating memory required will waste
space though.
`2exp' functions
It's up to an application to call functions like `mpz_mul_2exp'
when appropriate. General purpose functions like `mpz_mul' make
no attempt to identify powers of two or other special forms,
because such inputs will usually be very rare and testing every
time would be wasteful.
`ui' and `si' functions
The `ui' functions and the small number of `si' functions exist for
convenience and should be used where applicable. But if for
example an `mpz_t' contains a value that fits in an `unsigned
long' there's no need extract it and call a `ui' function, just
use the regular `mpz' function.
In-Place Operations
`mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
`mpf_neg' are fast when used for in-place operations like
`mpz_abs(x,x)', since in the current implementation only a single
field of `x' needs changing. On suitable compilers (GCC for
instance) this is inlined too.
`mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
from an in-place operation like `mpz_add_ui(x,x,y)', since usually
only one or two limbs of `x' will need to be changed. The same
applies to the full precision `mpz_add' etc if `y' is small. If
`y' is big then cache locality may be helped, but that's all.
`mpz_mul' is currently the opposite, a separate destination is
slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
only one limb, make a temporary copy of `x' before forming the
result. Normally that copying will only be a tiny fraction of the
time for the multiply, so this is not a particularly important
consideration.
`mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
attempt to recognise a copy of something to itself, so a call like
`mpz_set(x,x)' will be wasteful. Naturally that would never be
written deliberately, but if it might arise from two pointers to
the same object then a test to avoid it might be desirable.
if (x != y)
mpz_set (x, y);
Note that it's never worth introducing extra `mpz_set' calls just
to get in-place operations. If a result should go to a particular
variable then just direct it there and let GMP take care of data
movement.
Divisibility Testing (Small Integers)
`mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
functions for testing whether an `mpz_t' is divisible by an
individual small integer. They use an algorithm which is faster
than `mpz_tdiv_ui', but which gives no useful information about
the actual remainder, only whether it's zero (or a particular
value).
However when testing divisibility by several small integers, it's
best to take a remainder modulo their product, to save
multi-precision operations. For instance to test whether a number
is divisible by any of 23, 29 or 31 take a remainder modulo
23*29*31 = 20677 and then test that.
The division functions like `mpz_tdiv_q_ui' which give a quotient
as well as a remainder are generally a little slower than the
remainder-only functions like `mpz_tdiv_ui'. If the quotient is
only rarely wanted then it's probably best to just take a
remainder and then go back and calculate the quotient if and when
it's wanted (`mpz_divexact_ui' can be used if the remainder is
zero).
Rational Arithmetic
The `mpq' functions operate on `mpq_t' values with no common
factors in the numerator and denominator. Common factors are
checked-for and cast out as necessary. In general, cancelling
factors every time is the best approach since it minimizes the
sizes for subsequent operations.
However, applications that know something about the factorization
of the values they're working with might be able to avoid some of
the GCDs used for canonicalization, or swap them for divisions.
For example when multiplying by a prime it's enough to check for
factors of it in the denominator instead of doing a full GCD. Or
when forming a big product it might be known that very little
cancellation will be possible, and so canonicalization can be left
to the end.
The `mpq_numref' and `mpq_denref' macros give access to the
numerator and denominator to do things outside the scope of the
supplied `mpq' functions. *Note Applying Integer Functions::.
The canonical form for rationals allows mixed-type `mpq_t' and
integer additions or subtractions to be done directly with
multiples of the denominator. This will be somewhat faster than
`mpq_add'. For example,
/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
/* mpq += unsigned long */
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
/* mpq -= mpz */
mpz_submul (mpq_numref(q), mpq_denref(q), z);
Number Sequences
Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
designed for calculating isolated values. If a range of values is
wanted it's probably best to call to get a starting point and
iterate from there.
Text Input/Output
Hexadecimal or octal are suggested for input or output in text
form. Power-of-2 bases like these can be converted much more
efficiently than other bases, like decimal. For big numbers
there's usually nothing of particular interest to be seen in the
digits, so the base doesn't matter much.
Maybe we can hope octal will one day become the normal base for
everyday use, as proposed by King Charles XII of Sweden and later
reformers.
Debugging
=========
Stack Overflow
Depending on the system, a segmentation violation or bus error
might be the only indication of stack overflow. See
`--enable-alloca' choices in *Note Build Options::, for how to
address this.
Heap Problems
The most likely cause of application problems with GMP is heap
corruption. Failing to `init' GMP variables will have
unpredictable effects, and corruption arising elsewhere in a
program may well affect GMP. Initializing GMP variables more than
once or failing to clear them will cause memory leaks.
In all such cases a malloc debugger is recommended. On a GNU or
BSD system the standard C library `malloc' has some diagnostic
facilities, see *Note Allocation Debugging: (libc)Allocation
Debugging, or `man 3 malloc'. Other possibilities, in no
particular order, include
`http://www.inf.ethz.ch/personal/biere/projects/ccmalloc'
`http://quorum.tamu.edu/jon/gnu' (debauch)
`http://dmalloc.com'
`http://www.perens.com/FreeSoftware' (electric fence)
`http://packages.debian.org/fda'
`http://www.gnupdate.org/components/leakbug'
`http://people.redhat.com/~otaylor/memprof'
`http://www.cbmamiga.demon.co.uk/mpatrol'
The GMP default allocation routines in `memory.c' also have a
simple sentinel scheme which can be enabled with `#define DEBUG'
in that file. This is mainly designed for detecting buffer
overruns during GMP development, but might find other uses.
Stack Backtraces
On some systems the compiler options GMP uses by default can
interfere with debugging. In particular on x86 and 68k systems
`-fomit-frame-pointer' is used and this generally inhibits stack
backtracing. Recompiling without such options may help while
debugging, though the usual caveats about it potentially moving a
memory problem or hiding a compiler bug will apply.
GNU Debugger
A sample `.gdbinit' is included in the distribution, showing how
to call some undocumented dump functions to print GMP variables
from within GDB. Note that these functions shouldn't be used in
final application code since they're undocumented and may be
subject to incompatible changes in future versions of GMP.
Source File Paths
GMP has multiple source files with the same name, in different
directories. For example `mpz', `mpq', `mpf' and `mpfr' each have
an `init.c'. If the debugger can't already determine the right
one it may help to build with absolute paths on each C file. One
way to do that is to use a separate object directory with an
absolute path to the source directory.
cd /my/build/dir
/my/source/dir/gmp-4.1/configure
This works via `VPATH', and might require GNU `make'. Alternately
it might be possible to change the `.c.lo' rules appropriately.
Assertion Checking
The build option `--enable-assert' is available to add some
consistency checks to the library (see *Note Build Options::).
These are likely to be of limited value to most applications.
Assertion failures are just as likely to indicate memory
corruption as a library or compiler bug.
Applications using the low-level `mpn' functions, however, will
benefit from `--enable-assert' since it adds checks on the
parameters of most such functions, many of which have subtle
restrictions on their usage. Note however that only the generic C
code has checks, not the assembler code, so CPU `none' should be
used for maximum checking.
Temporary Memory Checking
The build option `--enable-alloca=debug' arranges that each block
of temporary memory in GMP is allocated with a separate call to
`malloc' (or the allocation function set with
`mp_set_memory_functions').
This can help a malloc debugger detect accesses outside the
intended bounds, or detect memory not released. In a normal
build, on the other hand, temporary memory is allocated in blocks
which GMP divides up for its own use, or may be allocated with a
compiler builtin `alloca' which will go nowhere near any malloc
debugger hooks.
Checker
The checker program (`http://savannah.gnu.org/projects/checker')
can be used with GMP. It contains a stub library which means GMP
applications compiled with checker can use a normal GMP build.
A build of GMP with checking within GMP itself can be made. This
will run very very slowly. Configure with
./configure --host=none-pc-linux-gnu CC=checkergcc
`--host=none' must be used, since the GMP assembler code doesn't
support the checking scheme. The GMP C++ features cannot be used,
since current versions of checker (0.9.9.1) don't yet support the
standard C++ library.
Valgrind
The valgrind program (`http://devel-home.kde.org/~sewardj') is a
memory checker for x86s. It translates and emulates machine
instructions to do strong checks for uninitialized data (at the
level of individual bits), memory accesses through bad pointers,
and memory leaks.
Current versions (20020226 snapshot) don't support MMX or SSE, so
GMP must be configured for an x86 without those (eg. plain
`i386'), or with a special `MPN_PATH' that excludes those
subdirectories (*note Build Options::).
Other Problems
Any suspected bug in GMP itself should be isolated to make sure
it's not an application problem, see *Note Reporting Bugs::.
Profiling
=========
Running a program under a profiler is a good way to find where it's
spending most time and where improvements can be best sought.
Depending on the system, it may be possible to get a flat profile,
meaning simple timer sampling of the program counter, with no special
GMP build options, just a `-p' when compiling the mainline. This is a
good way to ensure minimum interference with normal operation. The
necessary symbol type and size information exists in most of the GMP
assembler code.
The `--enable-profiling' build option can be used to add suitable
compiler flags, either for `prof' (`-p') or `gprof' (`-pg'), see *Note
Build Options::. Which of the two is available and what they do will
depend on the system, and possibly on support available in `libc'. For
some systems appropriate corresponding `mcount' calls are added to the
assembler code too.
On x86 systems `prof' gives call counting, so that average time spent
in a function can be determined. `gprof', where supported, adds call
graph construction, so for instance calls to `mpn_add_n' from `mpz_add'
and from `mpz_mul' can be differentiated.
On x86 and 68k systems `-pg' and `-fomit-frame-pointer' are
incompatible, so the latter is not used when `gprof' profiling is
selected, which may result in poorer code generation. If `prof'
profiling is selected instead it should still be possible to use
`gprof', but only the `gprof -p' flat profile and call counts can be
expected to be valid, not the `gprof -q' call graph.
Autoconf
========
Autoconf based applications can easily check whether GMP is
installed. The only thing to be noted is that GMP library symbols from
version 3 onwards have prefixes like `__gmpz'. The following therefore
would be a simple test,
AC_CHECK_LIB(gmp, __gmpz_init)
This just uses the default `AC_CHECK_LIB' actions for found or not
found, but an application that must have GMP would want to generate an
error if not found. For example,
AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR(
[GNU MP not found, see http://swox.com/gmp])])
If functions added in some particular version of GMP are required,
then one of those can be used when checking. For example `mpz_mul_si'
was added in GMP 3.1,
AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR(
[GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])
An alternative would be to test the version number in `gmp.h' using
say `AC_EGREP_CPP'. That would make it possible to test the exact
version, if some particular sub-minor release is known to be necessary.
An application that can use either GMP 2 or 3 will need to test for
`__gmpz_init' (GMP 3 and up) or `mpz_init' (GMP 2), and it's also worth
checking for `libgmp2' since Debian GNU/Linux systems used that name in
the past. For example,
AC_CHECK_LIB(gmp, __gmpz_init, ,
[AC_CHECK_LIB(gmp, mpz_init, ,
[AC_CHECK_LIB(gmp2, mpz_init)])])
In general it's suggested that applications should simply demand a
new enough GMP rather than trying to provide supplements for features
not available in past versions.
Occasionally an application will need or want to know the size of a
type at configuration or preprocessing time, not just with `sizeof' in
the code. This can be done in the normal way with `mp_limb_t' etc, but
GMP 4.0 or up is best for this, since prior versions needed certain
`-D' defines on systems using a `long long' limb. The following would
suit Autoconf 2.50 or up,
AC_CHECK_SIZEOF(mp_limb_t, , [#include ])
The optional `mpfr' functions are provided in a separate
`libmpfr.a', and this might be from GMP with `--enable-mpfr' or from
MPFR installed separately. Either way `libmpfr' depends on `libgmp',
it doesn't stand alone. Currently only a static `libmpfr.a' will be
available, not a shared library, since upward binary compatibility is
not guaranteed.
AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR(
[Need MPFR either from GNU MP 4 or separate MPFR package.
See http://www.mpfr.org or http://swox.com/gmp])
Reporting Bugs
**************
If you think you have found a bug in the GMP library, please
investigate it and report it. We have made this library available to
you, and it is not too much to ask you to report the bugs you find.
Before you report a bug, check it's not already addressed in *Note
Known Build Problems::, or perhaps *Note Notes for Particular
Systems::. You may also want to check `http://swox.com/gmp/' for
patches for this release.
Please include the following in any report,
* The GMP version number, and if pre-packaged or patched then say so.
* A test program that makes it possible for us to reproduce the bug.
Include instructions on how to run the program.
* A description of what is wrong. If the results are incorrect, in
what way. If you get a crash, say so.
* If you get a crash, include a stack backtrace from the debugger if
it's informative (`where' in `gdb', or `$C' in `adb').
* Please do not send core dumps, executables or `strace's.
* The configuration options you used when building GMP, if any.
* The name of the compiler and its version. For `gcc', get the
version with `gcc -v', otherwise perhaps `what `which cc`', or
similar.
* The output from running `uname -a'.
* The output from running `./config.guess', and from running
`./configfsf.guess' (might be the same).
* If the bug is related to `configure', then the contents of
`config.log'.
* If the bug is related to an `asm' file not assembling, then the
contents of `config.m4' and the offending line or lines from the
temporary `mpn/tmp-.s'.
Please make an effort to produce a self-contained report, with
something definite that can be tested or debugged. Vague queries or
piecemeal messages are difficult to act on and don't help the
development effort.
It is not uncommon that an observed problem is actually due to a bug
in the compiler; the GMP code tends to explore interesting corners in
compilers.
If your bug report is good, we will do our best to help you get a
corrected version of the library; if the bug report is poor, we won't
do anything about it (except maybe ask you to send a better report).
Send your report to: .
If you think something in this manual is unclear, or downright
incorrect, or if the language needs to be improved, please send a note
to the same address.
Integer Functions
*****************
This chapter describes the GMP functions for performing integer
arithmetic. These functions start with the prefix `mpz_'.
GMP integers are stored in objects of type `mpz_t'.
Initialization Functions
========================
The functions for integer arithmetic assume that all integer objects
are initialized. You do that by calling the function `mpz_init'. For
example,
{
mpz_t integ;
mpz_init (integ);
...
mpz_add (integ, ...);
...
mpz_sub (integ, ...);
/* Unless the program is about to exit, do ... */
mpz_clear (integ);
}
As you can see, you can store new values any number of times, once an
object is initialized.
- Function: void mpz_init (mpz_t INTEGER)
Initialize INTEGER, and set its value to 0.
- Function: void mpz_init2 (mpz_t INTEGER, unsigned long N)
Initialize INTEGER, with space for N bits, and set its value to 0.
N is only the initial space, INTEGER will grow automatically in
the normal way, if necessary, for subsequent values stored.
`mpz_init2' makes it possible to avoid such reallocations if a
maximum size is known in advance.
- Function: void mpz_clear (mpz_t INTEGER)
Free the space occupied by INTEGER. Call this function for all
`mpz_t' variables when you are done with them.
- Function: void mpz_realloc2 (mpz_t INTEGER, unsigned long N)
Change the space allocated for INTEGER to N bits. The value in
INTEGER is preserved if it fits, or is set to 0 if not.
This function can be used to increase the space for a variable in
order to avoid repeated automatic reallocations, or to decrease it
to give memory back to the heap.
- Function: void mpz_array_init (mpz_t INTEGER_ARRAY[], size_t
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
This is a special type of initialization. *Fixed* space of
FIXED_NUM_BITS bits is allocated to each of the ARRAY_SIZE
integers in INTEGER_ARRAY.
The space will not be automatically increased, unlike the normal
`mpz_init', but instead an application must ensure it's sufficient
for any value stored. The following space requirements apply to
various functions,
* `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
`mpz_set_ui' need room for the value they store.
* `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
for the larger of the two operands, plus an extra
`mp_bits_per_limb'.
* `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
of the number of bits in their operands, but each rounded up
to a multiple of `mp_bits_per_limb'.
* `mpz_swap' can be used between two array variables, but not
between an array and a normal variable.
For other functions, or if in doubt, the suggestion is to
calculate in a regular `mpz_init' variable and copy the result to
an array variable with `mpz_set'.
`mpz_array_init' can reduce memory usage in algorithms that need
large arrays of integers, since it avoids allocating and
reallocating lots of small memory blocks. There is no way to free
the storage allocated by this function. Don't call `mpz_clear'!
- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
Change the space for INTEGER to NEW_ALLOC limbs. The value in
INTEGER is preserved if it fits, or is set to 0 if not. The return
value is not useful to applications and should be ignored.
`mpz_realloc2' is the preferred way to accomplish allocation
changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
except that `_mpz_realloc' takes the new size in limbs.
Assignment Functions
====================
These functions assign new values to already initialized integers
(*note Initializing Integers::).
- Function: void mpz_set (mpz_t ROP, mpz_t OP)
- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
- Function: void mpz_set_si (mpz_t ROP, signed long int OP)
- Function: void mpz_set_d (mpz_t ROP, double OP)
- Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
- Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
Set the value of ROP from OP.
`mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
integer.
- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
Set the value of ROP from STR, a null-terminated C string in base
BASE. White space is allowed in the string, and is simply
ignored. The base may vary from 2 to 36. If BASE is 0, the
actual base is determined from the leading characters: if the
first two characters are "0x" or "0X", hexadecimal is assumed,
otherwise if the first character is "0", octal is assumed,
otherwise decimal is assumed.
This function returns 0 if the entire string is a valid number in
base BASE. Otherwise it returns -1.
[It turns out that it is not entirely true that this function
ignores white-space. It does ignore it between digits, but not
after a minus sign or within or after "0x". We are considering
changing the definition of this function, making it fail when
there is any white-space in the input, since that makes a lot of
sense. Send your opinion of this change to . Do
you really want it to accept "3 14" as meaning 314 as it does now?]
- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
Swap the values ROP1 and ROP2 efficiently.
Combined Initialization and Assignment Functions
================================================
For convenience, GMP provides a parallel series of
initialize-and-set functions which initialize the output and then store
the value there. These functions' names have the form `mpz_init_set...'
Here is an example of using one:
{
mpz_t pie;
mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
...
mpz_sub (pie, ...);
...
mpz_clear (pie);
}
Once the integer has been initialized by any of the `mpz_init_set...'
functions, it can be used as the source or destination operand for the
ordinary integer functions. Don't use an initialize-and-set function
on a variable already initialized!
- Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
- Function: void mpz_init_set_d (mpz_t ROP, double OP)
Initialize ROP with limb space and set the initial numeric value
from OP.
- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
Initialize ROP and set its value like `mpz_set_str' (see its
documentation above for details).
If the string is a correct base BASE number, the function returns
0; if an error occurs it returns -1. ROP is initialized even if
an error occurs. (I.e., you have to call `mpz_clear' for it.)
Conversion Functions
====================
This section describes functions for converting GMP integers to
standard C types. Functions for converting _to_ GMP integers are
described in *Note Assigning Integers:: and *Note I/O of Integers::.
- Function: unsigned long int mpz_get_ui (mpz_t OP)
Return the value of OP as an `unsigned long'.
If OP is too big to fit an `unsigned long' then just the least
significant bits that do fit are returned. The sign of OP is
ignored, only the absolute value is used.
- Function: signed long int mpz_get_si (mpz_t OP)
If OP fits into a `signed long int' return the value of OP.
Otherwise return the least significant part of OP, with the same
sign as OP.
If OP is too big to fit in a `signed long int', the returned
result is probably not very useful. To find out if the value will
fit, use the function `mpz_fits_slong_p'.
- Function: double mpz_get_d (mpz_t OP)
Convert OP to a `double'.
- Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP)
Find D and EXP such that D times 2 raised to EXP, with
0.5<=abs(D)<1, is a good approximation to OP.
- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
Convert OP to a string of digits in base BASE. The base may vary
from 2 to 36.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of storage large
enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
The two extra bytes are for a possible minus sign, and the
null-terminator.
A pointer to the result string is returned, being either the
allocated block, or the given STR.
- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
Return limb number N from OP. The sign of OP is ignored, just the
absolute value is used. The least significant limb is number 0.
`mpz_size' can be used to find how many limbs make up OP.
`mpz_getlimbn' returns zero if N is outside the range 0 to
`mpz_size(OP)-1'.
Arithmetic Functions
====================
- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
Set ROP to OP1 + OP2.
- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
OP2)
Set ROP to OP1 - OP2.
- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2)
- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
Set ROP to OP1 times OP2.
- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long
int OP2)
Set ROP to ROP + OP1 times OP2.
- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long
int OP2)
Set ROP to ROP - OP1 times OP2.
- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
Set ROP to OP1 times 2 raised to OP2. This operation can also be
defined as a left shift by OP2 bits.
- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
Set ROP to -OP.
- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
Set ROP to the absolute value of OP.
Division Functions
==================
Division is undefined if the divisor is zero. Passing a zero
divisor to the division or modulo functions (including the modular
powering functions `mpz_powm' and `mpz_powm_ui'), will cause an
intentional division by zero. This lets a program handle arithmetic
exceptions in these functions the same way as for normal C `int'
arithmetic.
- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
mpz_t N, unsigned long int D)
- Function: unsigned long int mpz_cdiv_ui (mpz_t N,
unsigned long int D)
- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N,
unsigned long int B)
- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N,
unsigned long int B)
- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
mpz_t N, unsigned long int D)
- Function: unsigned long int mpz_fdiv_ui (mpz_t N,
unsigned long int D)
- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N,
unsigned long int B)
- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N,
unsigned long int B)
- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
mpz_t N, unsigned long int D)
- Function: unsigned long int mpz_tdiv_ui (mpz_t N,
unsigned long int D)
- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N,
unsigned long int B)
- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N,
unsigned long int B)
Divide N by D, forming a quotient Q and/or remainder R. For the
`2exp' functions, D=2^B. The rounding is in three styles, each
suiting different applications.
* `cdiv' rounds Q up towards +infinity, and R will have the
opposite sign to D. The `c' stands for "ceil".
* `fdiv' rounds Q down towards -infinity, and R will have the
same sign as D. The `f' stands for "floor".
* `tdiv' rounds Q towards zero, and R will have the same sign
as N. The `t' stands for "truncate".
In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
0<=abs(R)1, such that OP equals A raised to the
power B.
Under this definition both 0 and 1 are considered to be perfect
powers. Negative values of OP are accepted, but of course can
only be odd perfect powers.
- Function: int mpz_perfect_square_p (mpz_t OP)
Return non-zero if OP is a perfect square, i.e., if the square
root of OP is an integer. Under this definition both 0 and 1 are
considered to be perfect squares.
Number Theoretic Functions
==========================
- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
Determine whether N is prime. Return 2 if N is definitely prime,
return 1 if N is probably prime (without being certain), or return
0 if N is definitely composite.
This function does some trial divisions, then some Miller-Rabin
probabilistic primality tests. REPS controls how many such tests
are done, 5 to 10 is a reasonable number, more will reduce the
chances of a composite being returned as "probably prime".
Miller-Rabin and similar tests can be more properly called
compositeness tests. Numbers which fail are known to be composite
but those which pass might be prime or might be composite. Only a
few composites pass, hence those which pass are considered
probably prime.
- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
Set ROP to the next prime greater than OP.
This function uses a probabilistic algorithm to identify primes.
For practical purposes it's adequate, the chance of a composite
passing will be extremely small.
- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to the greatest common divisor of OP1 and OP2. The result
is always positive even if one or both input operands are negative.
- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
unsigned long int OP2)
Compute the greatest common divisor of OP1 and OP2. If ROP is not
`NULL', store the result there.
If the result is small enough to fit in an `unsigned long int', it
is returned. If the result does not fit, 0 is returned, and the
result is equal to the argument OP1. Note that the result will
always fit if OP2 is non-zero.
- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t
B)
Set G to the greatest common divisor of A and B, and in addition
set S and T to coefficients satisfying A*S + B*T = G. G is always
positive, even if one or both of A and B are negative.
If T is `NULL' then that value is not computed.
- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2)
Set ROP to the least common multiple of OP1 and OP2. ROP is
always positive, irrespective of the signs of OP1 and OP2. ROP
will be zero if either OP1 or OP2 is zero.
- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Compute the inverse of OP1 modulo OP2 and put the result in ROP.
If the inverse exists, the return value is non-zero and ROP will
satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return
value is zero and ROP is undefined.
- Function: int mpz_jacobi (mpz_t A, mpz_t B)
Calculate the Jacobi symbol (A/B). This is defined only for B odd.
- Function: int mpz_legendre (mpz_t A, mpz_t P)
Calculate the Legendre symbol (A/P). This is defined only for P
an odd positive prime, and for such P it's identical to the Jacobi
symbol.
- Function: int mpz_kronecker (mpz_t A, mpz_t B)
- Function: int mpz_kronecker_si (mpz_t A, long B)
- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
- Function: int mpz_si_kronecker (long A, mpz_t B)
- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
Calculate the Jacobi symbol (A/B) with the Kronecker extension
(a/2)=(2/a) when a odd, or (a/2)=0 when a even.
When B is odd the Jacobi symbol and Kronecker symbol are
identical, so `mpz_kronecker_ui' etc can be used for mixed
precision Jacobi symbols too.
For more information see Henri Cohen section 1.4.2 (*note
References::), or any number theory textbook. See also the
example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
- Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
Remove all occurrences of the factor F from OP and store the
result in ROP. Return the multiplicity of F in OP.
- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
Set ROP to OP!, the factorial of OP.
- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
unsigned long int K)
Compute the binomial coefficient N over K and store the result in
ROP. Negative values of N are supported by `mpz_bin_ui', using
the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
section 1.2.6 part G.
- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
int N)
`mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
`mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
These functions are designed for calculating isolated Fibonacci
numbers. When a sequence of values is wanted it's best to start
with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
similar.
- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned long
int N)
`mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
`mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
These functions are designed for calculating isolated Lucas
numbers. When a sequence of values is wanted it's best to start
with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
or similar.
The Fibonacci numbers and Lucas numbers are related sequences, so
it's never necessary to call both `mpz_fib2_ui' and
`mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
can be found in *Note Lucas Numbers Algorithm::, the reverse is
straightforward too.
Comparison Functions
====================
- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
- Function: int mpz_cmp_d (mpz_t OP1, double OP2)
- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, or a negative value if OP1 < OP2.
Note that `mpz_cmp_ui' and `mpz_cmp_si' are macros and will
evaluate their arguments more than once.
- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
Compare the absolute values of OP1 and OP2. Return a positive
value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
negative value if abs(OP1) < abs(OP2).
Note that `mpz_cmpabs_si' is a macro and will evaluate its
arguments more than once.
- Macro: int mpz_sgn (mpz_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its argument multiple times.
Logical and Bit Manipulation Functions
======================================
These functions behave as if twos complement arithmetic were used
(although sign-magnitude is the actual implementation). The least
significant bit is number 0.
- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 logical-and OP2.
- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 inclusive-or OP2.
- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 exclusive-or OP2.
- Function: void mpz_com (mpz_t ROP, mpz_t OP)
Set ROP to the one's complement of OP.
- Function: unsigned long int mpz_popcount (mpz_t OP)
If OP>=0, return the population count of OP, which is the number
of 1 bits in the binary representation. If OP<0, the number of 1s
is infinite, and the return value is MAX_ULONG, the largest
possible `unsigned long'.
- Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
If OP1 and OP2 are both >=0 or both <0, return the hamming
distance between the two operands, which is the number of bit
positions where OP1 and OP2 have different bit values. If one
operand is >=0 and the other <0 then the number of bits different
is infinite, and the return value is MAX_ULONG, the largest
possible `unsigned long'.
- Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int
STARTING_BIT)
- Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int
STARTING_BIT)
Scan OP, starting from bit STARTING_BIT, towards more significant
bits, until the first 0 or 1 bit (respectively) is found. Return
the index of the found bit.
If the bit at STARTING_BIT is already what's sought, then
STARTING_BIT is returned.
If there's no bit found, then MAX_ULONG is returned. This will
happen in `mpz_scan0' past the end of a positive number, or
`mpz_scan1' past the end of a negative.
- Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)
Set bit BIT_INDEX in ROP.
- Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
Clear bit BIT_INDEX in ROP.
- Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
Input and Output Functions
==========================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream. Passing a `NULL' pointer for a STREAM
argument to any of these functions will make them read from `stdin' and
write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
Output OP on stdio stream STREAM, as a string of digits in base
BASE. The base may vary from 2 to 36.
Return the number of bytes written, or if an error occurred,
return 0.
- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
Input a possibly white-space preceded string in base BASE from
stdio stream STREAM, and put the read integer in ROP. The base
may vary from 2 to 36. If BASE is 0, the actual base is
determined from the leading characters: if the first two
characters are `0x' or `0X', hexadecimal is assumed, otherwise if
the first character is `0', octal is assumed, otherwise decimal is
assumed.
Return the number of bytes read, or if an error occurred, return 0.
- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
Output OP on stdio stream STREAM, in raw binary format. The
integer is written in a portable format, with 4 bytes of size
information, and that many bytes of limbs. Both the size and the
limbs are written in decreasing significance order (i.e., in
big-endian).
The output can be read with `mpz_inp_raw'.
Return the number of bytes written, or if an error occurred,
return 0.
The output of this can not be read by `mpz_inp_raw' from GMP 1,
because of changes necessary for compatibility between 32-bit and
64-bit machines.
- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
Input from stdio stream STREAM in the format written by
`mpz_out_raw', and put the result in ROP. Return the number of
bytes read, or if an error occurred, return 0.
This routine can read the output from `mpz_out_raw' also from GMP
1, in spite of changes necessary for compatibility between 32-bit
and 64-bit machines.
Random Number Functions
=======================
The random number functions of GMP come in two groups; older function
that rely on a global state, and newer functions that accept a state
parameter that is read and modified. Please see the *Note Random
Number Functions:: for more information on how to use and not to use
random number functions.
- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
unsigned long int N)
Generate a uniformly distributed random integer in the range 0 to
2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE, mpz_t
N)
Generate a uniform random integer in the range 0 to N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
unsigned long int N)
Generate a random integer with long strings of zeros and ones in
the binary representation. Useful for testing functions and
algorithms, since this kind of random numbers have proven to be
more likely to trigger corner-case bugs. The random number will
be in the range 0 to 2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs. The generated
random number doesn't satisfy any particular requirements of
randomness. Negative random numbers are generated when MAX_SIZE
is negative.
This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
instead.
- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. Useful
for testing functions and algorithms, since this kind of random
numbers have proven to be more likely to trigger corner-case bugs.
Negative random numbers are generated when MAX_SIZE is negative.
This function is obsolete. Use `mpz_rrandomb' instead.
Integer Import and Export
=========================
`mpz_t' variables can be converted to and from arbitrary words of
binary data with the following functions.
- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER, int
SIZE, int ENDIAN, size_t NAILS, const void *OP)
Set ROP from an array of word data at OP.
The parameters specify the format of the data. COUNT many words
are read, each SIZE bytes. ORDER can be 1 for most significant
word first or -1 for least significant first. Within each word
ENDIAN can be 1 for most significant byte first, -1 for least
significant first, or 0 for the native endianness of the host CPU.
The most significant NAILS bits of each word are skipped, this
can be 0 to use the full words.
There are no data alignment restrictions on OP, any address is
allowed.
Here's an example converting an array of `unsigned long' data, most
significant element first and host byte order within each value.
unsigned long a[20];
mpz_t z;
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
This example assumes the full `sizeof' bytes are used for data in
the given type, which is usually true, and certainly true for
`unsigned long' everywhere we know of. However on Cray vector
systems it may be noted that `short' and `int' are always stored
in 8 bytes (and with `sizeof' indicating that) but use only 32 or
46 bits. The NAILS feature can account for this, by passing for
instance `8*sizeof(int)-INT_BIT'.
- Function: void *mpz_export (void *ROP, size_t *COUNT, int ORDER, int
SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
Fill ROP with word data from OP.
The parameters specify the format of the data produced. Each word
will be SIZE bytes and ORDER can be 1 for most significant word
first or -1 for least significant first. Within each word ENDIAN
can be 1 for most significant byte first, -1 for least significant
first, or 0 for the native endianness of the host CPU. The most
significant NAILS bits of each word are unused and set to zero,
this can be 0 to produce full words.
The number of words produced is written to `*COUNT'. ROP must
have enough space for the data, or if ROP is `NULL' then a result
array of the necessary size is allocated using the current GMP
allocation function (*note Custom Allocation::). In either case
the return value is the destination used, ROP or the allocated
block.
If OP is non-zero then the most significant word produced will be
non-zero. If OP is zero then the count returned will be zero and
nothing written to ROP. If ROP is `NULL' in this case, no block
is allocated, just `NULL' is returned.
There are no data alignment restrictions on ROP, any address is
allowed. The sign of OP is ignored, just the absolute value is
used.
When an application is allocating space itself the required size
can be determined with a calculation like the following. Since
`mpz_sizeinbase' always returns at least 1, `count' here will be
at least one, which avoids any portability problems with
`malloc(0)', though if `z' is zero no space at all is actually
needed.
numb = 8*size - nail;
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
p = malloc (count * size);
Miscellaneous Functions
=======================
- Function: int mpz_fits_ulong_p (mpz_t OP)
- Function: int mpz_fits_slong_p (mpz_t OP)
- Function: int mpz_fits_uint_p (mpz_t OP)
- Function: int mpz_fits_sint_p (mpz_t OP)
- Function: int mpz_fits_ushort_p (mpz_t OP)
- Function: int mpz_fits_sshort_p (mpz_t OP)
Return non-zero iff the value of OP fits in an `unsigned long int',
`signed long int', `unsigned int', `signed int', `unsigned short
int', or `signed short int', respectively. Otherwise, return zero.
- Macro: int mpz_odd_p (mpz_t OP)
- Macro: int mpz_even_p (mpz_t OP)
Determine whether OP is odd or even, respectively. Return
non-zero if yes, zero if no. These macros evaluate their argument
more than once.
- Function: size_t mpz_size (mpz_t OP)
Return the size of OP measured in number of limbs. If OP is zero,
the returned value will be zero.
- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
Return the size of OP measured in number of digits in base BASE.
The base may vary from 2 to 36. The sign of OP is ignored, just
the absolute value is used. The result will be exact or 1 too
big. If BASE is a power of 2, the result will always be exact.
If OP is zero the return value is always 1.
This function is useful in order to allocate the right amount of
space before converting OP to a string. The right amount of
allocation is normally two more than the value returned by
`mpz_sizeinbase' (one extra for a minus sign and one for the
null-terminator).
Rational Number Functions
*************************
This chapter describes the GMP functions for performing arithmetic
on rational numbers. These functions start with the prefix `mpq_'.
Rational numbers are stored in objects of type `mpq_t'.
All rational arithmetic functions assume operands have a canonical
form, and canonicalize their result. The canonical from means that the
denominator and the numerator have no common factors, and that the
denominator is positive. Zero has the unique representation 0/1.
Pure assignment functions do not canonicalize the assigned variable.
It is the responsibility of the user to canonicalize the assigned
variable before any arithmetic operations are performed on that
variable.
- Function: void mpq_canonicalize (mpq_t OP)
Remove any factors that are common to the numerator and
denominator of OP, and make the denominator positive.
Initialization and Assignment Functions
=======================================
- Function: void mpq_init (mpq_t DEST_RATIONAL)
Initialize DEST_RATIONAL and set it to 0/1. Each variable should
normally only be initialized once, or at least cleared out (using
the function `mpq_clear') between each initialization.
- Function: void mpq_clear (mpq_t RATIONAL_NUMBER)
Free the space occupied by RATIONAL_NUMBER. Make sure to call this
function for all `mpq_t' variables when you are done with them.
- Function: void mpq_set (mpq_t ROP, mpq_t OP)
- Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
Assign ROP from OP.
- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
unsigned long int OP2)
- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
long int OP2)
Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
common factors, ROP has to be passed to `mpq_canonicalize' before
any operations are performed on ROP.
- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
Set ROP from a null-terminated string STR in the given BASE.
The string can be an integer like "41" or a fraction like
"41/152". The fraction must be in canonical form (*note Rational
Number Functions::), or if not then `mpq_canonicalize' must be
called.
The numerator and optional denominator are parsed the same as in
`mpz_set_str' (*note Assigning Integers::). White space is
allowed in the string, and is simply ignored. The BASE can vary
from 2 to 36, or if BASE is 0 then the leading characters are
used: `0x' for hex, `0' for octal, or decimal otherwise. Note
that this is done separately for the numerator and denominator, so
for instance `0xEF/100' is 239/100, whereas `0xEF/0x100' is
239/256.
The return value is 0 if the entire string is a valid number, or
-1 if not.
- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
Swap the values ROP1 and ROP2 efficiently.
Conversion Functions
====================
- Function: double mpq_get_d (mpq_t OP)
Convert OP to a `double'.
- Function: void mpq_set_d (mpq_t ROP, double OP)
- Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
Set ROP to the value of OP, without rounding.
- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
Convert OP to a string of digits in base BASE. The base may vary
from 2 to 36. The string will be of the form `num/den', or if the
denominator is 1 then just `num'.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of storage large
enough for the result, that being
mpz_sizeinbase (mpq_numref(OP), BASE)
+ mpz_sizeinbase (mpq_denref(OP), BASE) + 3
The three extra bytes are for a possible minus sign, possible
slash, and the null-terminator.
A pointer to the result string is returned, being either the
allocated block, or the given STR.
Arithmetic Functions
====================
- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
Set SUM to ADDEND1 + ADDEND2.
- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
SUBTRAHEND)
Set DIFFERENCE to MINUEND - SUBTRAHEND.
- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
MULTIPLICAND)
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, unsigned long int
OP2)
Set ROP to OP1 times 2 raised to OP2.
- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
DIVISOR)
Set QUOTIENT to DIVIDEND/DIVISOR.
- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, unsigned long int
OP2)
Set ROP to OP1 divided by 2 raised to OP2.
- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
Set NEGATED_OPERAND to -OPERAND.
- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
Set ROP to the absolute value of OP.
- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
this routine will divide by zero.
Comparison Functions
====================
- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
To determine if two rationals are equal, `mpq_equal' is faster than
`mpq_cmp'.
- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
long int DEN2)
- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
DEN2)
Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
NUM2/DEN2.
NUM2 and DEN2 are allowed to have common factors.
These functions are implemented as a macros and evaluate their
arguments multiple times.
- Macro: int mpq_sgn (mpq_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates its
arguments multiple times.
- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
Return non-zero if OP1 and OP2 are equal, zero if they are
non-equal. Although `mpq_cmp' can be used for the same purpose,
this function is much faster.
Applying Integer Functions to Rationals
=======================================
The set of `mpq' functions is quite small. In particular, there are
few functions for either input or output. The following functions give
direct access to the numerator and denominator of an `mpq_t'.
Note that if an assignment to the numerator and/or denominator could
take an `mpq_t' out of the canonical form described at the start of
this chapter (*note Rational Number Functions::) then
`mpq_canonicalize' must be called before any other `mpq' functions are
applied to that `mpq_t'.
- Macro: mpz_t mpq_numref (mpq_t OP)
- Macro: mpz_t mpq_denref (mpq_t OP)
Return a reference to the numerator and denominator of OP,
respectively. The `mpz' functions can be used on the result of
these macros.
- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
Get or set the numerator or denominator of a rational. These
functions are equivalent to calling `mpz_set' with an appropriate
`mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
`mpq_denref' is recommended instead of these functions.
Input and Output Functions
==========================
When using any of these functions, it's a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
Passing a `NULL' pointer for a STREAM argument to any of these
functions will make them read from `stdin' and write to `stdout',
respectively.
- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
Output OP on stdio stream STREAM, as a string of digits in base
BASE. The base may vary from 2 to 36. Output is in the form
`num/den' or if the denominator is 1 then just `num'.
Return the number of bytes written, or if an error occurred,
return 0.
- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
Read a string of digits from STREAM and convert them to a rational
in ROP. Any initial white-space characters are read and
discarded. Return the number of characters read (including white
space), or 0 if a rational could not be read.
The input can be a fraction like `17/63' or just an integer like
`123'. Reading stops at the first character not in this form, and
white space is not permitted within the string. If the input
might not be in canonical form, then `mpq_canonicalize' must be
called (*note Rational Number Functions::).
The BASE can be between 2 and 36, or can be 0 in which case the
leading characters of the string determine the base, `0x' or `0X'
for hexadecimal, `0' for octal, or decimal otherwise. The leading
characters are examined separately for the numerator and
denominator of a fraction, so for instance `0x10/11' is 16/11,
whereas `0x10/0x11' is 16/17.
Floating-point Functions
************************
GMP floating point numbers are stored in objects of type `mpf_t' and
functions operating on them have an `mpf_' prefix.
The mantissa of each float has a user-selectable precision, limited
only by available memory. Each variable has its own precision, and
that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on
most systems. In the current implementation the exponent is a count of
limbs, so for example on a 32-bit system this means a range of roughly
2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
greater. Note however `mpf_get_str' can only return an exponent which
fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
bigger than a `long'.
Each variable keeps a size for the mantissa data actually in use.
This means that if a float is exactly represented in only a few bits
then only those bits will be used in a calculation, even if the
selected precision is high.
All calculations are performed to the precision of the destination
variable. Each function is defined to calculate with "infinite
precision" followed by a truncation to the destination precision, but
of course the work done is only what's needed to determine a result
under that definition.
The precision selected for a variable is a minimum value, GMP may
increase it a little to facilitate efficient calculation. Currently
this means rounding up to a whole limb, and then sometimes having a
further partial limb, depending on the high limb of the mantissa. But
applications shouldn't be concerned by such details.
`mpf' functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the
exponent or results will be unpredictable. This might change in a
future release.
Note that the `mpf' functions are _not_ intended as a smooth
extension to IEEE P754 arithmetic. In particular results obtained on
one computer often differ from the results on a computer with a
different word size.
Initialization Functions
========================
- Function: void mpf_set_default_prec (unsigned long int PREC)
Set the default precision to be *at least* PREC bits. All
subsequent calls to `mpf_init' will use this precision, but
previously initialized variables are unaffected.
- Function: unsigned long int mpf_get_default_prec (void)
Return the default default precision actually used.
An `mpf_t' object must be initialized before storing the first value
in it. The functions `mpf_init' and `mpf_init2' are used for that
purpose.
- Function: void mpf_init (mpf_t X)
Initialize X to 0. Normally, a variable should be initialized
once only or at least be cleared, using `mpf_clear', between
initializations. The precision of X is undefined unless a default
precision has already been established by a call to
`mpf_set_default_prec'.
- Function: void mpf_init2 (mpf_t X, unsigned long int PREC)
Initialize X to 0 and set its precision to be *at least* PREC
bits. Normally, a variable should be initialized once only or at
least be cleared, using `mpf_clear', between initializations.
- Function: void mpf_clear (mpf_t X)
Free the space occupied by X. Make sure to call this function for
all `mpf_t' variables when you are done with them.
Here is an example on how to initialize floating-point variables:
{
mpf_t x, y;
mpf_init (x); /* use default precision */
mpf_init2 (y, 256); /* precision _at least_ 256 bits */
...
/* Unless the program is about to exit, do ... */
mpf_clear (x);
mpf_clear (y);
}
The following three functions are useful for changing the precision
during a calculation. A typical use would be for adjusting the
precision gradually in iterative algorithms like Newton-Raphson, making
the computation precision closely match the actual accurate part of the
numbers.
- Function: unsigned long int mpf_get_prec (mpf_t OP)
Return the current precision of OP, in bits.
- Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC)
Set the precision of ROP to be *at least* PREC bits. The value in
ROP will be truncated to the new precision.
This function requires a call to `realloc', and so should not be
used in a tight loop.
- Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC)
Set the precision of ROP to be *at least* PREC bits, without
changing the memory allocated.
PREC must be no more than the allocated precision for ROP, that
being the precision when ROP was initialized, or in the most recent
`mpf_set_prec'.
The value in ROP is unchanged, and in particular if it had a higher
precision than PREC it will retain that higher precision. New
values written to ROP will use the new PREC.
Before calling `mpf_clear' or the full `mpf_set_prec', another
`mpf_set_prec_raw' call must be made to restore ROP to its original
allocated precision. Failing to do so will have unpredictable
results.
`mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
original allocated precision. After `mpf_set_prec_raw' it
reflects the PREC value set.
`mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
at different precisions during a calculation, perhaps to gradually
increase precision in an iteration, or just to use various
different precisions for different purposes during a calculation.
Assignment Functions
====================
These functions assign new values to already initialized floats
(*note Initializing Floats::).
- Function: void mpf_set (mpf_t ROP, mpf_t OP)
- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
- Function: void mpf_set_d (mpf_t ROP, double OP)
- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
Set the value of ROP from OP.
- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
Set the value of ROP from the string in STR. The string is of the
form `M@N' or, if the base is 10 or less, alternatively `MeN'.
`M' is the mantissa and `N' is the exponent. The mantissa is
always in the specified base. The exponent is either in the
specified base or, if BASE is negative, in decimal. The decimal
point expected is taken from the current locale, on systems
providing `localeconv'.
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
Negative values are used to specify that the exponent is in
decimal.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
White space is allowed in the string, and is simply ignored.
[This is not really true; white-space is ignored in the beginning
of the string and within the mantissa, but not in other places,
such as after a minus sign or in the exponent. We are considering
changing the definition of this function, making it fail when
there is any white-space in the input, since that makes a lot of
sense. Please tell us your opinion about this change. Do you
really want it to accept "3 14" as meaning 314 as it does now?]
This function returns 0 if the entire string is a valid number in
base BASE. Otherwise it returns -1.
- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
Swap ROP1 and ROP2 efficiently. Both the values and the
precisions of the two variables are swapped.
Combined Initialization and Assignment Functions
================================================
For convenience, GMP provides a parallel series of
initialize-and-set functions which initialize the output and then store
the value there. These functions' names have the form `mpf_init_set...'
Once the float has been initialized by any of the `mpf_init_set...'
functions, it can be used as the source or destination operand for the
ordinary float functions. Don't use an initialize-and-set function on
a variable already initialized!
- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
- Function: void mpf_init_set_d (mpf_t ROP, double OP)
Initialize ROP and set its value from OP.
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.
- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
Initialize ROP and set its value from the string in STR. See
`mpf_set_str' above for details on the assignment operation.
Note that ROP is initialized even if an error occurs. (I.e., you
have to call `mpf_clear' for it.)
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.
Conversion Functions
====================
- Function: double mpf_get_d (mpf_t OP)
Convert OP to a `double'.
- Function: double mpf_get_d_2exp (signed long int EXP, mpf_t OP)
Find D and EXP such that D times 2 raised to EXP, with
0.5<=abs(D)<1, is a good approximation to OP. This is similar to
the standard C function `frexp'.
- Function: long mpf_get_si (mpf_t OP)
- Function: unsigned long mpf_get_ui (mpf_t OP)
Convert OP to a `long' or `unsigned long', truncating any fraction
part. If OP is too big for the return type, the result is
undefined.
See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
Miscellaneous Float Functions::).
- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE,
size_t N_DIGITS, mpf_t OP)
Convert OP to a string of digits in base BASE. BASE can be 2 to
36. Up to N_DIGITS digits will be generated. Trailing zeros are
not returned. No more digits than can be accurately represented
by OP are ever generated. If N_DIGITS is 0 then that accurate
maximum number of digits are generated.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of N\_DIGITS + 2
bytes, that being enough for the mantissa, a possible minus sign,
and a null-terminator. When N_DIGITS is 0 to get all significant
digits, an application won't be able to know the space required,
and STR should be `NULL' in that case.
The generated string is a fraction, with an implicit radix point
immediately to the left of the first digit. The applicable
exponent is written through the EXPPTR pointer. For example, the
number 3.1416 would be returned as string "31416" and exponent 1.
When OP is zero, an empty string is produced and the exponent
returned is 0.
A pointer to the result string is returned, being either the
allocated block or the given STR.
Arithmetic Functions
====================
- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 + OP2.
- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
OP2)
- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 - OP2.
- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 times OP2.
Division is undefined if the divisor is zero, and passing a zero
divisor to the divide functions will make these functions intentionally
divide by zero. This lets the user handle arithmetic exceptions in
these functions in the same manner as other arithmetic exceptions.
- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
OP2)
- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1/OP2.
- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
Set ROP to the square root of OP.
- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 raised to the power OP2.
- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
Set ROP to -OP.
- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
Set ROP to the absolute value of OP.
- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 times 2 raised to OP2.
- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 divided by 2 raised to OP2.
Comparison Functions
====================
- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3)
Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
zero otherwise. I.e., test of OP1 and OP2 are approximately equal.
Caution: Currently only whole limbs are compared, and only in an
exact fashion. In the future values like 1000 and 0111 may be
considered the same to 3 bits (on the basis that their difference
is that small).
- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
Compute the relative difference between OP1 and OP2 and store the
result in ROP. This is abs(OP1-OP2)/OP1.
- Macro: int mpf_sgn (mpf_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its arguments multiple times.
Input and Output Functions
==========================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream. Passing a `NULL' pointer for a STREAM
argument to any of these functions will make them read from `stdin' and
write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
N_DIGITS, mpf_t OP)
Print OP to STREAM, as a string of digits. Return the number of
bytes written, or if an error occurred, return 0.
The mantissa is prefixed with an `0.' and is in the given BASE,
which may vary from 2 to 36. An exponent then printed, separated
by an `e', or if BASE is greater than 10 then by an `@'. The
exponent is always in decimal. The decimal point follows the
current locale, on systems providing `localeconv'.
Up to N_DIGITS will be printed from the mantissa, except that no
more digits than are accurately representable by OP will be
printed. N_DIGITS can be 0 to select that accurate maximum.
- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
Read a string in base BASE from STREAM, and put the read float in
ROP. The string is of the form `M@N' or, if the base is 10 or
less, alternatively `MeN'. `M' is the mantissa and `N' is the
exponent. The mantissa is always in the specified base. The
exponent is either in the specified base or, if BASE is negative,
in decimal. The decimal point expected is taken from the current
locale, on systems providing `localeconv'.
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
Negative values are used to specify that the exponent is in
decimal.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
Return the number of bytes read, or if an error occurred, return 0.
Miscellaneous Functions
=======================
- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
to the integer towards zero.
- Function: int mpf_integer_p (mpf_t OP)
Return non-zero if OP is an integer.
- Function: int mpf_fits_ulong_p (mpf_t OP)
- Function: int mpf_fits_slong_p (mpf_t OP)
- Function: int mpf_fits_uint_p (mpf_t OP)
- Function: int mpf_fits_sint_p (mpf_t OP)
- Function: int mpf_fits_ushort_p (mpf_t OP)
- Function: int mpf_fits_sshort_p (mpf_t OP)
Return non-zero if OP would fit in the respective C data type, when
truncated to an integer.
- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
unsigned long int NBITS)
Generate a uniformly distributed random float in ROP, such that 0
<= ROP < 1, with NBITS significant bits in the mantissa.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
EXP)
Generate a random float of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. The
exponent of the number is in the interval -EXP to EXP. This
function is useful for testing functions and algorithms, since
this kind of random numbers have proven to be more likely to
trigger corner-case bugs. Negative random numbers are generated
when MAX_SIZE is negative.
Low-level Functions
*******************
This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.
These functions start with the prefix `mpn_'.
The `mpn' functions are designed to be as fast as possible, *not* to
provide a coherent calling interface. The different functions have
somewhat similar interfaces, but there are variations that make them
hard to use. These functions do as little as possible apart from the
real multiple precision computation, so that no time is spent on things
that not all callers need.
A source operand is specified by a pointer to the least significant
limb and a limb count. A destination operand is specified by just a
pointer. It is the responsibility of the caller to ensure that the
destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform
computations on subranges of an argument, and store the result into a
subrange of a destination.
A common requirement for all functions is that each source area
needs at least one limb. No size argument may be zero. Unless
otherwise stated, in-place operations are allowed where source and
destination are the same, but not where they only partly overlap.
The `mpn' functions are the base for the implementation of the
`mpz_', `mpf_', and `mpq_' functions.
This example adds the number beginning at S1P and the number
beginning at S2P and writes the sum at DESTP. All areas have N limbs.
cy = mpn_add_n (destp, s1p, s2p, n)
In the notation used here, a source operand is identified by the
pointer to the least significant limb, and the limb count in braces.
For example, {S1P, S1N}.
- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
This is the lowest-level function for addition. It is the
preferred function for addition, since it is written in assembly
for most CPUs. For addition of a variable to itself (i.e., S1P
equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.
- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Add {S1P, N} and S2LIMB, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
limbs of the result to RP. Return carry, either 0 or 1.
This function requires that S1N is greater than or equal to S2N.
- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Subtract {S2P, N} from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
This is the lowest-level function for subtraction. It is the
preferred function for subtraction, since it is written in
assembly for most CPUs.
- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Subtract S2LIMB from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
significant limbs of the result to RP. Return borrow, either 0 or
1.
This function requires that S1N is greater than or equal to S2N.
- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const
mp_limb_t *S2P, mp_size_t N)
Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
RP.
The destination has to have space for 2*N limbs, even if the
product's most significant limb is zero.
- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} by S2LIMB, and write the N least significant
limbs of the product to RP. Return the most significant limb of
the product. {S1P, N} and {RP, N} are allowed to overlap provided
RP <= S1P.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written
in assembly for most CPUs.
Don't call this function if S2LIMB is a power of 2; use
`mpn_lshift' with a count equal to the logarithm of S2LIMB
instead, for optimal speed.
- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and add the N least significant
limbs of the product to {RP, N} and write the result to RP.
Return the most significant limb of the product, plus carry-out
from the addition.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written
in assembly for most CPUs.
- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and subtract the N least significant
limbs of the product from {RP, N} and write the result to RP.
Return the most significant limb of the product, minus borrow-out
from the subtraction.
This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP.
It is written in assembly for most CPUs.
- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Multiply {S1P, S1N} and {S2P, S2N}, and write the result to RP.
Return the most significant limb of the result.
The destination has to have space for S1N + S2N limbs, even if the
result might be one limb smaller.
This function requires that S1N is greater than or equal to S2N.
The destination must be distinct from both input operands.
- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
mp_size_t DN)
Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
and the remainder at {RP, DN}. The quotient is rounded towards 0.
No overlap is permitted between arguments. NN must be greater
than or equal to DN. The most significant limb of DP must be
non-zero. The QXN operand must be zero.
- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
mp_size_t S3N)
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
best performance.]
Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
with the exception of the most significant limb, which is
returned. The remainder replaces the dividend at RS2P; it will be
S3N limbs long (i.e., as many limbs as the divisor).
In addition to an integer quotient, QXN fraction limbs are
developed, and stored after the integral limbs. For most usages,
QXN will be zero.
It is required that RS2N is greater than or equal to S3N. It is
required that the most significant bit of the divisor is set.
If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
that special case, no overlap between arguments is permitted.
Return the most significant limb of the quotient, either 0 or 1.
The area at R1P needs to be RS2N - S3N + QXN limbs large.
- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
mp_size_t S2N, mp_limb_t S3LIMB)
Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
Return the remainder.
The integer quotient is written to {R1P+QXN, S2N} and in addition
QXN fraction limbs are developed and written to {R1P, QXN}.
Either or both S2N and QXN can be zero. For most usages, QXN will
be zero.
`mpn_divmod_1' exists for upward source compatibility and is
simply a macro calling `mpn_divrem_1' with a QXN of 0.
The areas at R1P and S2P have to be identical or completely
separate, not partially overlapping.
- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
best performance.]
- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t N)
- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t N, mp_limb_t CARRY)
Divide {SP, N} by 3, expecting it to divide exactly, and writing
the result to {RP, N}. If 3 divides exactly, the return value is
zero and the result is the quotient. If not, the return value is
non-zero and the result won't be anything useful.
`mpn_divexact_by3c' takes an initial carry parameter, which can be
the return value from a previous call, so a large calculation can
be done piece by piece from low to high. `mpn_divexact_by3' is
simply a macro calling `mpn_divexact_by3c' with a 0 carry
parameter.
These routines use a multiply-by-inverse and will be faster than
`mpn_divrem_1' on CPUs with fast multiplication but slow division.
The source a, result q, size n, initial carry i, and return value
c satisfy c*b^n + a-i = 3*q, where b=2^mp_bits_per_limb. The
return c is always 0, 1 or 2, and the initial carry i must also be
0, 1 or 2 (these are both borrows really). When c=0 clearly
q=(a-i)/3. When c!=0, the remainder (a-i) mod 3 is given by 3-c,
because b == 1 mod 3 (when `mp_bits_per_limb' is even, which is
always so currently).
- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
zero.
- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, unsigned
long int D)
This function puts the low floor(D/mp_bits_per_limb) limbs of Q =
{S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod
`mp_bits_per_limb' bits of Q.
{S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed
at S1P. Since the low floor(D/mp_bits_per_limb) limbs of this
difference are zero, it is possible to overwrite the low limbs at
S1P with this difference, provided RP <= S1P.
This function requires that S1N * mp_bits_per_limb >= D, and that
{S2P, S2N} is odd.
*This interface is preliminary. It might change incompatibly in
future revisions.*
- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsigned int COUNT)
Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
The bits shifted out at the left are returned in the least
significant COUNT bits of the return value (the rest of the return
value is zero).
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
{SP, N} and {RP, N} may overlap, provided RP >= SP.
This function is written in assembly for most CPUs.
- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsigned int COUNT)
Shift {SP, N} right by COUNT bits, and write the result to {RP,
N}. The bits shifted out at the right are returned in the most
significant COUNT bits of the return value (the rest of the return
value is zero).
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
{SP, N} and {RP, N} may overlap, provided RP <= SP.
This function is written in assembly for most CPUs.
- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
mp_size_t N)
Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
S2, 0 if they are equal, or a negative value if S1 < S2.
- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N)
Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and
{S2P, S2N}. The result can be up to S2N limbs, the return value
is the actual number produced. Both source operands are destroyed.
{S1P, S1N} must have at least as many bits as {S2P, S2N}. {S2P,
S2N} must be odd. Both operands must have non-zero most
significant limbs. No overlap is permitted between {S1P, S1N} and
{S2P, S2N}.
- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Return the greatest common divisor of {S1P, S1N} and S2LIMB. Both
operands must be non-zero.
- Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,
mp_size_t *R2N, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t
*S2P, mp_size_t S2N)
Calculate the greatest common divisor of {S1P, S1N} and {S2P,
S2N}. Store the gcd at {R1P, RETVAL} and the first cofactor at
{R2P, *R2N}, with *R2N negative if the cofactor is negative. R1P
and R2P should each have room for S1N+1 limbs, but the return
value and value stored through R2N indicate the actual number
produced.
{S1P, S1N} >= {S2P, S2N} is required, and both must be non-zero.
The regions {S1P, S1N+1} and {S2P, S2N+1} are destroyed (i.e. the
operands plus an extra limb past the end of each).
The cofactor R1 will satisfy R2*S1 + K*S2 = R1. The second
cofactor K is not calculated but can easily be obtained from (R1 -
R2*S1) / S2.
- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
const mp_limb_t *SP, mp_size_t N)
Compute the square root of {SP, N} and put the result at {R1P,
ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space
for N limbs, but the return value indicates how many are produced.
The most significant limb of {SP, N} must be non-zero. The areas
{R1P, ceil(N/2)} and {SP, N} must be completely separate. The
areas {R2P, N} and {SP, N} must be either identical or completely
separate.
If the remainder is not wanted then R2P can be `NULL', and in this
case the return value is zero or non-zero according to whether the
remainder would have been zero or non-zero.
A return value of zero indicates a perfect square. See also
`mpz_perfect_square_p'.
- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
mp_limb_t *S1P, mp_size_t S1N)
Convert {S1P, S1N} to a raw unsigned char array at STR in base
BASE, and return the number of characters produced. There may be
leading zeros in the string. The string is not in ASCII; to
convert it to printable format, add the ASCII codes for `0' or
`A', depending on the base and range.
The most significant limb of the input {S1P, S1N} must be
non-zero. The input {S1P, S1N} is clobbered, except when BASE is
a power of 2, in which case it's unchanged.
The area at STR has to have space for the largest possible number
represented by a S1N long limb array, plus one extra character.
- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const char *STR,
size_t STRSIZE, int BASE)
Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
STR[0] is the most significant byte and STR[STRSIZE-1] is the
least significant. Each byte should be a value in the range 0 to
BASE-1, not an ASCII character. BASE can vary from 2 to 256.
The return value is the number of limbs written to RP. If the most
significant input byte is non-zero then the high limb at RP will be
non-zero, and only that exact number of limbs will be required
there.
If the most significant input byte is zero then there may be high
zero limbs written to RP and included in the return value.
STRSIZE must be at least 1, and no overlap is permitted between
{STR,STRSIZE} and the result at RP.
- Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,
unsigned long int BIT)
Scan S1P from bit position BIT for the next clear bit.
It is required that there be a clear bit within the area at S1P at
or beyond bit position BIT, so that the function has something to
return.
- Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P,
unsigned long int BIT)
Scan S1P from bit position BIT for the next set bit.
It is required that there be a set bit within the area at S1P at or
beyond bit position BIT, so that the function has something to
return.
- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
Generate a random number of length R1N and store it at R1P. The
most significant limb is always non-zero. `mpn_random' generates
uniformly distributed limb data, `mpn_random2' generates long
strings of zeros and ones in the binary representation.
`mpn_random2' is intended for testing the correctness of the `mpn'
routines.
- Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,
mp_size_t N)
Count the number of set bits in {S1P, N}.
- Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const
mp_limb_t *S2P, mp_size_t N)
Compute the hamming distance between {S1P, N} and {S2P, N}.
- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
N)
Return non-zero iff {S1P, N} is a perfect square.
Nails
=====
*Everything in this section is highly experimental and may disappear
or be subject to incompatible changes in a future version of GMP.*
Nails are an experimental feature whereby a few bits are left unused
at the top of each `mp_limb_t'. This can significantly improve carry
handling on some processors.
All the `mpn' functions accepting limb data will expect the nail
bits to be zero on entry, and will return data with the nails similarly
all zero. This applies both to limb vectors and to single limb
arguments.
Nails can be enabled by configuring with `--enable-nails'. By
default the number of bits will be chosen according to what suits the
host processor, but a particular number can be selected with
`--enable-nails=N'.
At the mpn level, a nail build is neither source nor binary
compatible with a non-nail build, strictly speaking. But programs
acting on limbs only through the mpn functions are likely to work
equally well with either build, and judicious use of the definitions
below should make any program compatible with either build, at the
source level.
For the higher level routines, meaning `mpz' etc, a nail build
should be fully source and binary compatible with a non-nail build.
- Macro: GMP_NAIL_BITS
- Macro: GMP_NUMB_BITS
- Macro: GMP_LIMB_BITS
`GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
not in use. `GMP_NUMB_BITS' is the number of data bits in a limb.
`GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In
all cases
GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
- Macro: GMP_NAIL_MASK
- Macro: GMP_NUMB_MASK
Bit masks for the nail and number parts of a limb.
`GMP_NAIL_MASK' is 0 when nails are not in use.
`GMP_NAIL_MASK' is not often needed, since the nail part can be
obtained with `x >> GMP_NUMB_BITS', and that means one less large
constant, which can help various RISC chips.
- Macro: GMP_NUMB_MAX
The maximum value that can be stored in the number part of a limb.
This is the same as `GMP_NUMB_MASK', but can be used for clarity
when doing comparisons rather than bit-wise operations.
The term "nails" comes from finger or toe nails, which are at the
ends of a limb (arm or leg). "numb" is short for number, but is also
how the developers felt after trying for a long time to come up with
sensible names for these things.
In the future (the distant future most likely) a non-zero nail might
be permitted, giving non-unique representations for numbers in a limb
vector. This would help vector processors since carries would only
ever need to propagate one or two limbs.
Random Number Functions
***********************
Sequences of pseudo-random numbers in GMP are generated using a
variable of type `gmp_randstate_t', which holds an algorithm selection
and a current state. Such a variable must be initialized by a call to
one of the `gmp_randinit' functions, and can be seeded with one of the
`gmp_randseed' functions.
The functions actually generating random numbers are described in
*Note Integer Random Numbers::, and *Note Miscellaneous Float
Functions::.
The older style random number functions don't accept a
`gmp_randstate_t' parameter but instead share a global variable of that
type. They use a default algorithm and are currently not seeded
(though perhaps that will change in the future). The new functions
accepting a `gmp_randstate_t' are recommended for applications that
care about randomness.
Random State Initialization
===========================
- Function: void gmp_randinit_default (gmp_randstate_t STATE)
Initialize STATE with a default algorithm. This will be a
compromise between speed and randomness, and is recommended for
applications with no special requirements.
- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A,
unsigned long C, unsigned long M2EXP)
Initialize STATE with a linear congruential algorithm X = (A*X +
C) mod 2^M2EXP.
The low bits of X in this algorithm are not very random. The least
significant bit will have a period no more than 2, and the second
bit no more than 4, etc. For this reason only the high half of
each X is actually used.
When a random number of more than M2EXP/2 bits is to be generated,
multiple iterations of the recurrence are used and the results
concatenated.
- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
unsigned long SIZE)
Initialize STATE for a linear congruential algorithm as per
`gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
chosen so that SIZE bits (or more) of each X will be used, ie.
M2EXP >= SIZE/2.
If successful the return value is non-zero. If SIZE is bigger
than the table data provides then the return value is zero. The
maximum SIZE currently supported is 128.
- Function: void gmp_randinit (gmp_randstate_t STATE,
gmp_randalg_t ALG, ...)
*This function is obsolete.*
Initialize STATE with an algorithm selected by ALG. The only
choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size'.
A third parameter of type `unsigned long' is required, this is the
SIZE for that function. `GMP_RAND_ALG_DEFAULT' or 0 are the same
as `GMP_RAND_ALG_LC'.
`gmp_randinit' sets bits in `gmp_errno' to indicate an error.
`GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is unsupported, or
`GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter is too big.
- Function: void gmp_randclear (gmp_randstate_t STATE)
Free all memory occupied by STATE.
Random State Seeding
====================
- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
unsigned long int SEED)
Set an initial seed value into STATE.
The size of a seed determines how many different sequences of
random numbers that it's possible to generate. The "quality" of
the seed is the randomness of a given seed compared to the
previous seed used, and this affects the randomness of separate
number sequences. The method for choosing a seed is critical if
the generated numbers are to be used for important applications,
such as generating cryptographic keys.
Traditionally the system time has been used to seed, but care
needs to be taken with this. If an application seeds often and
the resolution of the system clock is low, then the same sequence
of numbers might be repeated. Also, the system time is quite easy
to guess, so if unpredictability is required then it should
definitely not be the only source for the seed value. On some
systems there's a special device `/dev/random' which provides
random data better suited for use as a seed.
Formatted Output
****************
Format Strings
==============
`gmp_printf' and friends accept format strings similar to the
standard C `printf' (*note Formatted Output: (libc)Formatted Output.).
A format specification is of the form
% [flags] [width] [.[precision]] [type] conv
GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
respectively, and `N' for an `mp_limb_t' array. `Z', `Q' and `N'
behave like integers. `Q' will print a `/' and a denominator, if
needed. `F' behaves like a float. For example,
mpz_t z;
gmp_printf ("%s is an mpz %Zd\n", "here", z);
mpq_t q;
gmp_printf ("a hex rational: %#40Qx\n", q);
mpf_t f;
int n;
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
const mp_limb_t *ptr;
mp_size_t size;
gmp_printf ("limb array %Nx\n", ptr, size);
For `N' the limbs are expected least significant first, as per the
`mpn' functions (*note Low-level Functions::). A negative size can be
given to print the value as a negative.
All the standard C `printf' types behave the same as the C library
`printf', and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are
simply handed to `printf' and only the GMP extensions handled directly.
The flags accepted are as follows. GLIBC style ' is only for the
standard C types (not the GMP types), and only if the C library
supports it.
0 pad with zeros (rather than spaces)
# show the base with `0x', `0X' or `0'
+ always show a sign
(space) show a space or a `-' sign
' group digits, GLIBC style (not GMP types)
The optional width and precision can be given as a number within the
format string, or as a `*' to take an extra parameter of type `int', the
same as the standard `printf'.
The standard types accepted are as follows. `h' and `l' are
portable, the rest will depend on the compiler (or include files) for
the type and the C library for the output.
h short
hh char
j intmax_t or uintmax_t
l long or wchar_t
ll same as L
L long long or long double
q quad_t or u_quad_t
t ptrdiff_t
z size_t
The GMP types are
F mpf_t, float conversions
Q mpq_t, integer conversions
N mp_limb_t array, integer conversions
Z mpz_t, integer conversions
The conversions accepted are as follows. `a' and `A' are always
supported for `mpf_t' but depend on the C library for standard C float
types. `m' and `p' depend on the C library.
a A hex floats, GLIBC style
c character
d decimal integer
e E scientific format float
f fixed point float
i same as d
g G fixed or scientific float
m `strerror' string, GLIBC style
n store characters written so far
o octal integer
p pointer
s string
u unsigned integer
x X hex integer
`o', `x' and `X' are unsigned for the standard C types, but for
types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z',
`Q' and `N'.
`n' can be used with any type, even the GMP types.
Other types or conversions that might be accepted by the C library
`printf' cannot be used through `gmp_printf', this includes for
instance extensions registered with GLIBC `register_printf_function'.
Also currently there's no support for POSIX `$' style numbered arguments
(perhaps this will be added in the future).
The precision field has it's usual meaning for integer `Z' and float
`F' types, but is currently undefined for `Q' and should not be used
with that.
`mpf_t' conversions only ever generate as many digits as can be
accurately represented by the operand, the same as `mpf_get_str' does.
Zeros will be used if necessary to pad to the requested precision. This
happens even for an `f' conversion of an `mpf_t' which is an integer,
for instance 2^1024 in an `mpf_t' of 128 bits precision will only
produce about 20 digits, then pad with zeros to the decimal point. An
empty precision field like `%.Fe' or `%.Ff' can be used to specifically
request just the significant digits.
The decimal point character (or string) is taken from the current
locale settings on systems which provide `localeconv' (*note Locales
and Internationalization: (libc)Locales.). The C library will normally
do the same for standard float output.
Functions
=========
Each of the following functions is similar to the corresponding C
library function. The basic `printf' forms take a variable argument
list. The `vprintf' forms take an argument pointer, see *Note Variadic
Functions: (libc)Variadic Functions, or `man 3 va_start'.
It should be emphasised that if a format string is invalid, or the
arguments don't match what the format specifies, then the behaviour of
any of these functions will be unpredictable. GCC format string
checking is not available, since it doesn't recognise the GMP
extensions.
The file based functions `gmp_printf' and `gmp_fprintf' will return
-1 to indicate a write error. All the functions can return -1 if the C
library `printf' variant in use returns -1, but this shouldn't normally
occur.
- Function: int gmp_printf (const char *FMT, ...)
- Function: int gmp_vprintf (const char *FMT, va_list AP)
Print to the standard output `stdout'. Return the number of
characters written, or -1 if an error occurred.
- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
Print to the stream FP. Return the number of characters written,
or -1 if an error occurred.
- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
Form a null-terminated string in BUF. Return the number of
characters written, excluding the terminating null.
No overlap is permitted between the space at BUF and the string
FMT.
These functions are not recommended, since there's no protection
against exceeding the space available at BUF.
- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char *FMT,
...)
- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
*FMT, va_list AP)
Form a null-terminated string in BUF. No more than SIZE bytes
will be written. To get the full output, SIZE must be enough for
the string and null-terminator.
The return value is the total number of characters which ought to
have been produced, excluding the terminating null. If RETVAL >=
SIZE then the actual output has been truncated to the first SIZE-1
characters, and a null appended.
No overlap is permitted between the region {BUF,SIZE} and the FMT
string.
Notice the return value is in ISO C99 `snprintf' style. This is
so even if the C library `vsnprintf' is the older GLIBC 2.0.x
style.
- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
- Function: int gmp_vasprintf (char *PP, const char *FMT, va_list AP)
Form a null-terminated string in a block of memory obtained from
the current memory allocation function (*note Custom
Allocation::). The block will be the size of the string and
null-terminator. Put the address of the block in *PP. Return the
number of characters produced, excluding the null-terminator.
Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
if there's no more memory available, it lets the current allocation
function handle that.
- Function: int gmp_obstack_printf (struct obstack *OB, const char
*FMT, ...)
- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
*FMT, va_list AP)
Append to the current obstack object, in the same style as
`obstack_printf'. Return the number of characters written. A
null-terminator is not written.
FMT cannot be within the current obstack object, since the object
might move as it grows.
These functions are available only when the C library provides the
obstack feature, which probably means only on GNU systems, see
*Note Obstacks: (libc)Obstacks.
C++ Formatted Output
====================
The following functions are provided in `libgmpxx', which is built
if C++ support is enabled (*note Build Options::). Prototypes are
available from `'.
- Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
Print OP to STREAM, using its `ios' formatting settings.
`ios::width' is reset to 0 after output, the same as the standard
`ostream operator<<' routines do.
In hex or octal, OP is printed as a signed number, the same as for
decimal. This is unlike the standard `operator<<' routines on
`int' etc, which instead give twos complement.
- Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
Print OP to STREAM, using its `ios' formatting settings.
`ios::width' is reset to 0 after output, the same as the standard
`ostream operator<<' routines do.
Output will be a fraction like `5/9', or if the denominator is 1
then just a plain integer like `123'.
In hex or octal, OP is printed as a signed value, the same as for
decimal. If `ios::showbase' is set then a base indicator is shown
on both the numerator and denominator (if the denominator is
required).
- Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
Print OP to STREAM, using its `ios' formatting settings.
`ios::width' is reset to 0 after output, the same as the standard
`ostream operator<<' routines do. The decimal point follows the
current locale, on systems providing `localeconv'.
Hex and octal are supported, unlike the standard `operator<<' on
`double'. The mantissa will be in hex or octal, the exponent will
be in decimal. For hex the exponent delimiter is an `@'. This is
as per `mpf_out_str'.
`ios::showbase' is supported, and will put a base on the mantissa,
for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
This last form is slightly strange, but at least differentiates
itself from decimal.
These operators mean that GMP types can be printed in the usual C++
way, for example,
mpz_t z;
int n;
...
cout << "iteration " << n << " value " << z << "\n";
But note that `ostream' output (and `istream' input, *note C++
Formatted Input::) is the only overloading available and using for
instance `+' with an `mpz_t' will have unpredictable results.
Formatted Input
***************
Formatted Input Strings
=======================
`gmp_scanf' and friends accept format strings similar to the
standard C `scanf' (*note Formatted Input: (libc)Formatted Input.). A
format specification is of the form
% [flags] [width] [type] conv
GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
respectively. `Z' and `Q' behave like integers. `Q' will read a `/'
and a denominator, if present. `F' behaves like a float.
GMP variables don't require an `&' when passed to `gmp_scanf', since
they're already "call-by-reference". For example,
/* to read say "a(5) = 1234" */
int n;
mpz_t z;
gmp_scanf ("a(%d) = %Zd\n", &n, z);
mpq_t q1, q2;
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
/* to read say "topleft (1.55,-2.66)" */
mpf_t x, y;
char buf[32];
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
All the standard C `scanf' types behave the same as in the C library
`scanf', and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are
simply handed to `scanf' and only the GMP extensions handled directly.
The flags accepted are as follows. `a' and `'' will depend on
support from the C library, and `'' cannot be used with GMP types.
* read but don't store
a allocate a buffer (string conversions)
' group digits, GLIBC style (not GMP types)
The standard types accepted are as follows. `h' and `l' are
portable, the rest will depend on the compiler (or include files) for
the type and the C library for the input.
h short
hh char
j intmax_t or uintmax_t
l long or wchar_t
ll same as L
L long long or long double
q quad_t or u_quad_t
t ptrdiff_t
z size_t
The GMP types are
F mpf_t, float conversions
Q mpq_t, integer conversions
Z mpz_t, integer conversions
The conversions accepted are as follows. `p' and `[' will depend on
support from the C library, the rest are standard.
c character or characters
d decimal integer
e E f g G float
i integer with base indicator
n characters written so far
o octal integer
p pointer
s string of non-whitespace characters
u decimal integer
x X hex integer
[ string of characters in a set
`e', `E', `f', `g' and `G' are identical, they all read either fixed
point or scientific format, and either `e' or `E' for the exponent in
scientific format.
`x' and `X' are identical, both accept both upper and lower case
hexadecimal.
`o', `u', `x' and `X' all read positive or negative values. For the
standard C types these are described as "unsigned" conversions, but
that merely affects certain overflow handling, negatives are still
allowed (see `strtoul', *Note Parsing of Integers: (libc)Parsing of
Integers). For GMP types there are no overflows, and `d' and `u' are
identical.
`Q' type reads the numerator and (optional) denominator as given.
If the value might not be in canonical form then `mpq_canonicalize'
must be called before using it in any calculations (*note Rational
Number Functions::).
`Qi' will read a base specification separately for the numerator and
denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11'
would be 16/17.
`n' can be used with any of the types above, even the GMP types.
`*' to suppress assignment is allowed, though the field would then do
nothing at all.
Other conversions or types that might be accepted by the C library
`scanf' cannot be used through `gmp_scanf'.
Whitespace is read and discarded before a field, except for `c' and
`[' conversions.
For float conversions, the decimal point character (or string)
expected is taken from the current locale settings on systems which
provide `localeconv' (*note Locales and Internationalization:
(libc)Locales.). The C library will normally do the same for standard
float input.
Formatted Input Functions
=========================
Each of the following functions is similar to the corresponding C
library function. The plain `scanf' forms take a variable argument
list. The `vscanf' forms take an argument pointer, see *Note Variadic
Functions: (libc)Variadic Functions, or `man 3 va_start'.
It should be emphasised that if a format string is invalid, or the
arguments don't match what the format specifies, then the behaviour of
any of these functions will be unpredictable. GCC format string
checking is not available, since it doesn't recognise the GMP
extensions.
No overlap is permitted between the FMT string and any of the results
produced.
- Function: int gmp_scanf (const char *FMT, ...)
- Function: int gmp_vscanf (const char *FMT, va_list AP)
Read from the standard input `stdin'.
- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
Read from the stream FP.
- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
AP)
Read from a null-terminated string S.
The return value from each of these functions is the same as the
standard C99 `scanf', namely the number of fields successfully parsed
and stored. `%n' fields and fields read but suppressed by `*' don't
count towards the return value.
If end of file or file error, or end of string, is reached when a
match is required, and when no previous non-suppressed fields have
matched, then the return value is EOF instead of 0. A match is
required for a literal character in the format string or a field other
than `%n'. Whitespace in the format string is only an optional match
and won't induce an EOF in this fashion. Leading whitespace read and
discarded for a field doesn't count as a match.
C++ Formatted Input
===================
The following functions are provided in `libgmpxx', which is built
only if C++ support is enabled (*note Build Options::). Prototypes are
available from `'.
- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
Read ROP from STREAM, using its `ios' formatting settings.
- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
Read ROP from STREAM, using its `ios' formatting settings.
An integer like `123' will be read, or a fraction like `5/9'. If
the fraction is not in canonical form then `mpq_canonicalize' must
be called (*note Rational Number Functions::).
- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
Read ROP from STREAM, using its `ios' formatting settings.
Hex or octal floats are not supported, but might be in the future.
These operators mean that GMP types can be read in the usual C++
way, for example,
mpz_t z;
...
cin >> z;
But note that `istream' input (and `ostream' output, *note C++
Formatted Output::) is the only overloading available and using for
instance `+' with an `mpz_t' will have unpredictable results.
C++ Class Interface
*******************
This chapter describes the C++ class based interface to GMP.
All GMP C language types and functions can be used in C++ programs,
since `gmp.h' has `extern "C"' qualifiers, but the class interface
offers overloaded functions and operators which may be more convenient.
Due to the implementation of this interface, a reasonably recent C++
compiler is required, one supporting namespaces, partial specialization
of templates and member templates. For GCC this means version 2.91 or
later.
*Everything described in this chapter is to be considered preliminary
and might be subject to incompatible changes if some unforeseen
difficulty reveals itself.*
C++ Interface General
=====================
All the C++ classes and functions are available with
#include
The classes defined are
- Class: mpz_class
- Class: mpq_class
- Class: mpf_class
The standard operators and various standard functions are overloaded
to allow arithmetic with these classes. For example,
int
main (void)
{
mpz_class a, b, c;
a = 1234;
b = "-5678";
c = a+b;
cout << "sum is " << c << "\n";
cout << "absolute value is " << abs(c) << "\n";
return 0;
}
An important feature of the implementation is that an expression like
`a=b+c' results in a single call to the corresponding `mpz_add',
without using a temporary for the `b+c' part. Expressions which by
their nature imply intermediate values, like `a=b*c+d*e', still use
temporaries though.
The classes can be freely intermixed in expressions, as can the
classes and the standard types `long', `unsigned long' and `double'.
Smaller types like `int' or `float' can also be intermixed, since C++
will promote them.
Note that `bool' is not accepted directly, but must be explicitly
cast to an `int' first. This is because C++ will automatically convert
any pointer to a `bool', so if GMP accepted `bool' it would make all
sorts of invalid class and pointer combinations compile but almost
certainly not do anything sensible.
Conversions back from the classes to standard C++ types aren't done
automatically, instead member functions like `get_si' are provided (see
the following sections for details).
Also there are no automatic conversions from the classes to the
corresponding GMP C types, instead a reference to the underlying C
object can be obtained with the following functions,
- Function: mpz_t mpz_class::get_mpz_t ()
- Function: mpq_t mpq_class::get_mpq_t ()
- Function: mpf_t mpf_class::get_mpf_t ()
These can be used to call a C function which doesn't have a C++ class
interface. For example to set `a' to the GCD of `b' and `c',
mpz_class a, b, c;
...
mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
In the other direction, a class can be initialized from the
corresponding GMP C type, or assigned to if an explicit constructor is
used. In both cases this makes a copy of the value, it doesn't create
any sort of association. For example,
mpz_t z;
// ... init and calculate z ...
mpz_class x(z);
mpz_class y;
y = mpz_class (z);
There are no namespace setups in `gmpxx.h', all types and functions
are simply put into the global namespace. This is what `gmp.h' has
done in the past, and continues to do for compatibility. The extras
provided by `gmpxx.h' follow GMP naming conventions and are unlikely to
clash with anything.
C++ Interface Integers
======================
- Function: void mpz_class::mpz_class (type N)
Construct an `mpz_class'. All the standard C++ types may be used,
except `long long' and `long double', and all the GMP C++ classes
can be used. Any necessary conversion follows the corresponding C
function, for example `double' follows `mpz_set_d' (*note
Assigning Integers::).
- Function: void mpz_class::mpz_class (mpz_t Z)
Construct an `mpz_class' from an `mpz_t'. The value in Z is
copied into the new `mpz_class', there won't be any permanent
association between it and Z.
- Function: void mpz_class::mpz_class (const char *S)
- Function: void mpz_class::mpz_class (const char *S, int base)
- Function: void mpz_class::mpz_class (const string& S)
- Function: void mpz_class::mpz_class (const string& S, int base)
Construct an `mpz_class' converted from a string using
`mpz_set_str', (*note Assigning Integers::). If the BASE is not
given then 0 is used.
- Function: mpz_class operator/ (mpz_class A, mpz_class D)
- Function: mpz_class operator% (mpz_class A, mpz_class D)
Divisions involving `mpz_class' round towards zero, as per the
`mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::).
This corresponds to the rounding used for plain `int' calculations
on most machines.
The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called
directly if desired. For example,
mpz_class q, a, d;
...
mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
- Function: mpz_class abs (mpz_class OP1)
- Function: int cmp (mpz_class OP1, type OP2)
- Function: int cmp (type OP1, mpz_class OP2)
- Function: double mpz_class::get_d (void)
- Function: long mpz_class::get_si (void)
- Function: unsigned long mpz_class::get_ui (void)
- Function: bool mpz_class::fits_sint_p (void)
- Function: bool mpz_class::fits_slong_p (void)
- Function: bool mpz_class::fits_sshort_p (void)
- Function: bool mpz_class::fits_uint_p (void)
- Function: bool mpz_class::fits_ulong_p (void)
- Function: bool mpz_class::fits_ushort_p (void)
- Function: int sgn (mpz_class OP)
- Function: mpz_class sqrt (mpz_class OP)
These functions provide a C++ class interface to the corresponding
GMP C routines.
`cmp' can be used with any of the classes or the standard C++
types, except `long long' and `long double'.
Overloaded operators for combinations of `mpz_class' and `double'
are provided for completeness, but it should be noted that if the given
`double' is not an integer then the way any rounding is done is
currently unspecified. The rounding might take place at the start, in
the middle, or at the end of the operation, and it might change in the
future.
Conversions between `mpz_class' and `double', however, are defined
to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
And comparisons are always made exactly, as per `mpz_cmp_d'.
C++ Interface Rationals
=======================
In all the following constructors, if a fraction is given then it
should be in canonical form, or if not then `mpq_class::canonicalize'
called.
- Function: void mpq_class::mpq_class (type OP)
- Function: void mpq_class::mpq_class (integer NUM, integer DEN)
Construct an `mpq_class'. The initial value can be a single value
of any type, or a pair of integers (`mpz_class' or standard C++
integer types) representing a fraction, except that `long long'
and `long double' are not supported. For example,
mpq_class q (99);
mpq_class q (1.75);
mpq_class q (1, 3);
- Function: void mpq_class::mpq_class (mpq_t Q)
Construct an `mpq_class' from an `mpq_t'. The value in Q is
copied into the new `mpq_class', there won't be any permanent
association between it and Q.
- Function: void mpq_class::mpq_class (const char *S)
- Function: void mpq_class::mpq_class (const char *S, int base)
- Function: void mpq_class::mpq_class (const string& S)
- Function: void mpq_class::mpq_class (const string& S, int base)
Construct an `mpq_class' converted from a string using
`mpq_set_str', (*note Initializing Rationals::). If the BASE is
not given then 0 is used.
- Function: void mpq_class::canonicalize ()
Put an `mpq_class' into canonical form, as per *Note Rational
Number Functions::. All arithmetic operators require their
operands in canonical form, and will return results in canonical
form.
- Function: mpq_class abs (mpq_class OP)
- Function: int cmp (mpq_class OP1, type OP2)
- Function: int cmp (type OP1, mpq_class OP2)
- Function: double mpq_class::get_d (void)
- Function: int sgn (mpq_class OP)
These functions provide a C++ class interface to the corresponding
GMP C routines.
`cmp' can be used with any of the classes or the standard C++
types, except `long long' and `long double'.
- Function: mpz_class& mpq_class::get_num ()
- Function: mpz_class& mpq_class::get_den ()
Get a reference to an `mpz_class' which is the numerator or
denominator of an `mpq_class'. This can be used both for read and
write access. If the object returned is modified, it modifies the
original `mpq_class'.
If direct manipulation might produce a non-canonical value, then
`mpq_class::canonicalize' must be called before further operations.
- Function: mpz_t mpq_class::get_num_mpz_t ()
- Function: mpz_t mpq_class::get_den_mpz_t ()
Get a reference to the underlying `mpz_t' numerator or denominator
of an `mpq_class'. This can be passed to C functions expecting an
`mpz_t'. Any modifications made to the `mpz_t' will modify the
original `mpq_class'.
If direct manipulation might produce a non-canonical value, then
`mpq_class::canonicalize' must be called before further operations.
- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
Read ROP from STREAM, using its `ios' formatting settings, the
same as `mpq_t operator>>' (*note C++ Formatted Input::).
If the ROP read might not be in canonical form then
`mpq_class::canonicalize' must be called.
C++ Interface Floats
====================
When an expression requires the use of temporary intermediate
`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
same precision as the destination `f'. Explicit constructors can be
used if this doesn't suit.
- Function: mpf_class::mpf_class (type OP)
- Function: mpf_class::mpf_class (type OP, unsigned long PREC)
Construct an `mpf_class'. Any standard C++ type can be used,
except `long long' and `long double', and any of the GMP C++
classes can be used.
If PREC is given, the initial precision is that value, in bits. If
PREC is not given, then the initial precision is determined by the
type of OP given. An `mpz_class', `mpq_class', string, or C++
builtin type will give the default `mpf' precision (*note
Initializing Floats::). An `mpf_class' or expression will give
the precision of that value. The precision of a binary expression
is the higher of the two operands.
mpf_class f(1.5); // default precision
mpf_class f(1.5, 500); // 500 bits (at least)
mpf_class f(x); // precision of x
mpf_class f(abs(x)); // precision of x
mpf_class f(-g, 1000); // 1000 bits (at least)
mpf_class f(x+y); // greater of precisions of x and y
- Function: mpf_class abs (mpf_class OP)
- Function: mpf_class ceil (mpf_class OP)
- Function: int cmp (mpf_class OP1, type OP2)
- Function: int cmp (type OP1, mpf_class OP2)
- Function: mpf_class floor (mpf_class OP)
- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
- Function: double mpf_class::get_d (void)
- Function: long mpf_class::get_si (void)
- Function: unsigned long mpf_class::get_ui (void)
- Function: bool mpf_class::fits_sint_p (void)
- Function: bool mpf_class::fits_slong_p (void)
- Function: bool mpf_class::fits_sshort_p (void)
- Function: bool mpf_class::fits_uint_p (void)
- Function: bool mpf_class::fits_ulong_p (void)
- Function: bool mpf_class::fits_ushort_p (void)
- Function: int sgn (mpf_class OP)
- Function: mpf_class sqrt (mpf_class OP)
- Function: mpf_class trunc (mpf_class OP)
These functions provide a C++ class interface to the corresponding
GMP C routines.
`cmp' can be used with any of the classes or the standard C++
types, except `long long' and `long double'.
The accuracy provided by `hypot' is not currently guaranteed.
- Function: unsigned long int mpf_class::get_prec ()
- Function: void mpf_class::set_prec (unsigned long PREC)
- Function: void mpf_class::set_prec_raw (unsigned long PREC)
Get or set the current precision of an `mpf_class'.
The restrictions described for `mpf_set_prec_raw' (*note
Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note
in particular that the `mpf_class' must be restored to it's
allocated precision before being destroyed. This must be done by
application code, there's no automatic mechanism for it.
C++ Interface MPFR
==================
The C++ class interface to MPFR is provided if MPFR is enabled
(*note Build Options::). This interface must be regarded as
preliminary and possibly subject to incompatible changes in the future,
since MPFR itself is preliminary. All definitions can be obtained with
#include
This defines
- Class: mpfr_class
which behaves similarly to `mpf_class' (*note C++ Interface Floats::).
C++ Interface Random Numbers
============================
- Class: gmp_randclass
The C++ class interface to the GMP random number functions uses
`gmp_randclass' to hold an algorithm selection and current state,
as per `gmp_randstate_t'.
- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
(gmp_randstate_t, ...), ...)
Construct a `gmp_randclass', using a call to the given RANDINIT
function (*note Random State Initialization::). The arguments
expected are the same as RANDINIT, but with `mpz_class' instead of
`mpz_t'. For example,
gmp_randclass r1 (gmp_randinit_default);
gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
`gmp_randinit_lc_2exp_size' can fail if the size requested is too
big, the behaviour of `gmp_randclass::gmp_randclass' is undefined
in this case (perhaps this will change in the future).
- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
Construct a `gmp_randclass' using the same parameters as
`gmp_randinit' (*note Random State Initialization::). This
function is obsolete and the above RANDINIT style should be
preferred.
- Function: void gmp_randclass::seed (unsigned long int S)
- Function: void gmp_randclass::seed (mpz_class S)
Seed a random number generator. See *note Random Number
Functions::, for how to choose a good seed.
- Function: mpz_class gmp_randclass::get_z_bits (unsigned long BITS)
- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
Generate a random integer with a specified number of bits.
- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
Generate a random integer in the range 0 to N-1 inclusive.
- Function: mpf_class gmp_randclass::get_f ()
- Function: mpf_class gmp_randclass::get_f (unsigned long PREC)
Generate a random float F in the range 0 <= F < 1. F will be to
PREC bits precision, or if PREC is not given then to the precision
of the destination. For example,
gmp_randclass r;
...
mpf_class f (0, 512); // 512 bits precision
f = r.get_f(); // random number, 512 bits
C++ Interface Limitations
=========================
`mpq_class' and Templated Reading
A generic piece of template code probably won't know that
`mpq_class' requires a `canonicalize' call if inputs read with
`operator>>' might be non-canonical. This can lead to incorrect
results.
`operator>>' behaves as it does for reasons of efficiency. A
canonicalize can be quite time consuming on large operands, and is
best avoided if it's not necessary.
But this potential difficulty reduces the usefulness of
`mpq_class'. Perhaps a mechanism to tell `operator>>' what to do
will be adopted in the future, maybe a preprocessor define, a
global flag, or an `ios' flag pressed into service. Or maybe, at
the risk of inconsistency, the `mpq_class' `operator>>' could
canonicalize and leave `mpq_t' `operator>>' not doing so, for use
on those occasions when that's acceptable. Send feedback or
alternate ideas to .
Subclassing
Subclassing the GMP C++ classes works, but is not currently
recommended.
Expressions involving subclasses resolve correctly (or seem to),
but in normal C++ fashion the subclass doesn't inherit
constructors and assignments. There's many of those in the GMP
classes, and a good way to reestablish them in a subclass is not
yet provided.
Templated Expressions
A subtle difficulty exists when using expressions together with
application-defined template functions. Consider the following,
with `T' intended to be some numeric type,
template
T fun (const T &, const T &);
When used with, say, plain `mpz_class' variables, it works fine:
`T' is resolved as `mpz_class'.
mpz_class f(1), g(2);
fun (f, g); // Good
But when one of the arguments is an expression, it doesn't work.
mpz_class f(1), g(2), h(3);
fun (f, g+h); // Bad
This is because `g+h' ends up being a certain expression template
type internal to `gmpxx.h', which the C++ template resolution
rules are unable to automatically convert to `mpz_class'. The
workaround is simply to add an explicit cast.
mpz_class f(1), g(2), h(3);
fun (f, mpz_class(g+h)); // Good
Similarly, within `fun' it may be necessary to cast an expression
to type `T' when calling a templated `fun2'.
template
void fun (T f, T g)
{
fun2 (f, f+g); // Bad
}
template
void fun (T f, T g)
{
fun2 (f, T(f+g)); // Good
}
Berkeley MP Compatible Functions
********************************
These functions are intended to be fully compatible with the
Berkeley MP library which is available on many BSD derived U*ix
systems. The `--enable-mpbsd' option must be used when building GNU MP
to make these available (*note Installing GMP::).
The original Berkeley MP library has a usage restriction: you cannot
use the same variable as both source and destination in a single
function call. The compatible functions in GNU MP do not share this
restriction--inputs and outputs may overlap.
It is not recommended that new programs are written using these
functions. Apart from the incomplete set of functions, the interface
for initializing `MINT' objects is more error prone, and the `pow'
function collides with `pow' in `libm.a'.
Include the header `mp.h' to get the definition of the necessary
types and functions. If you are on a BSD derived system, make sure to
include GNU `mp.h' if you are going to link the GNU `libmp.a' to your
program. This means that you probably need to give the `-I'
option to the compiler, where `' is the directory where you have
GNU `mp.h'.
- Function: MINT * itom (signed short int INITIAL_VALUE)
Allocate an integer consisting of a `MINT' object and dynamic limb
space. Initialize the integer to INITIAL_VALUE. Return a pointer
to the `MINT' object.
- Function: MINT * xtom (char *INITIAL_VALUE)
Allocate an integer consisting of a `MINT' object and dynamic limb
space. Initialize the integer from INITIAL_VALUE, a hexadecimal,
null-terminated C string. Return a pointer to the `MINT' object.
- Function: void move (MINT *SRC, MINT *DEST)
Set DEST to SRC by copying. Both variables must be previously
initialized.
- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
Add SRC_1 and SRC_2 and put the sum in DESTINATION.
- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.
- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
Multiply SRC_1 and SRC_2 and put the product in DESTINATION.
- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,
MINT *REMAINDER)
- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT
*QUOTIENT, signed short int *REMAINDER)
Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod
DIVISOR. The quotient is rounded towards zero; the remainder has
the same sign as the dividend unless it is zero.
Some implementations of these functions work differently--or not
at all--for negative arguments.
- Function: void msqrt (MINT *OP, MINT *ROOT, MINT *REMAINDER)
Set ROOT to the truncated integer part of the square root of OP,
like `mpz_sqrt'. Set REMAINDER to OP-ROOT*ROOT, i.e. zero if OP
is a perfect square.
If ROOT and REMAINDER are the same variable, the results are
undefined.
- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
Set DEST to (BASE raised to EXP) modulo MOD.
- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
Set DEST to BASE raised to EXP.
- Function: void gcd (MINT *OP1, MINT *OP2, MINT *RES)
Set RES to the greatest common divisor of OP1 and OP2.
- Function: int mcmp (MINT *OP1, MINT *OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
- Function: void min (MINT *DEST)
Input a decimal string from `stdin', and put the read integer in
DEST. SPC and TAB are allowed in the number string, and are
ignored.
- Function: void mout (MINT *SRC)
Output SRC to `stdout', as a decimal string. Also output a
newline.
- Function: char * mtox (MINT *OP)
Convert OP to a hexadecimal string, and return a pointer to the
string. The returned string is allocated using the default memory
allocation function, `malloc' by default. It will be
`strlen(str)+1' bytes, that being exactly enough for the string
and null-terminator.
- Function: void mfree (MINT *OP)
De-allocate, the space used by OP. *This function should only be
passed a value returned by `itom' or `xtom'.*
Custom Allocation
*****************
By default GMP uses `malloc', `realloc' and `free' for memory
allocation, and if they fail GMP prints a message to the standard error
output and terminates the program.
Alternate functions can be specified to allocate memory in a
different way or to have a different error action on running out of
memory.
This feature is available in the Berkeley compatibility library
(*note BSD Compatible Functions::) as well as the main GMP library.
- Function: void mp_set_memory_functions (
void *(*ALLOC_FUNC_PTR) (size_t),
void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
void (*FREE_FUNC_PTR) (void *, size_t))
Replace the current allocation functions from the arguments. If
an argument is `NULL', the corresponding default function is used.
These functions will be used for all memory allocation done by
GMP, apart from temporary space from `alloca' if that function is
available and GMP is configured to use it (*note Build Options::).
*Be sure to call `mp_set_memory_functions' only when there are no
active GMP objects allocated using the previous memory functions!
Usually that means calling it before any other GMP function.*
The functions supplied should fit the following declarations:
- Function: void * allocate_function (size_t ALLOC_SIZE)
Return a pointer to newly allocated space with at least ALLOC_SIZE
bytes.
- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
size_t NEW_SIZE)
Resize a previously allocated block PTR of OLD_SIZE bytes to be
NEW_SIZE bytes.
The block may be moved if necessary or if desired, and in that
case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
the new location. The return value is a pointer to the resized
block, that being the new location if moved or just PTR if not.
PTR is never `NULL', it's always a previously allocated block.
NEW_SIZE may be bigger or smaller than OLD_SIZE.
- Function: void deallocate_function (void *PTR, size_t SIZE)
De-allocate the space pointed to by PTR.
PTR is never `NULL', it's always a previously allocated block of
SIZE bytes.
A "byte" here means the unit used by the `sizeof' operator.
The OLD_SIZE parameters to REALLOCATE_FUNCTION and
DEALLOCATE_FUNCTION are passed for convenience, but of course can be
ignored if not needed. The default functions using `malloc' and friends
for instance don't use them.
No error return is allowed from any of these functions, if they
return then they must have performed the specified operation. In
particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
return `NULL'.
Getting a different fatal error action is a good use for custom
allocation functions, for example giving a graphical dialog rather than
the default print to `stderr'. How much is possible when genuinely out
of memory is another question though.
There's currently no defined way for the allocation functions to
recover from an error such as out of memory, they must terminate
program execution. A `longjmp' or throwing a C++ exception will have
undefined results. This may change in the future.
GMP may use allocated blocks to hold pointers to other allocated
blocks. This will limit the assumptions a conservative garbage
collection scheme can make.
Since the default GMP allocation uses `malloc' and friends, those
functions will be linked in even if the first thing a program does is an
`mp_set_memory_functions'. It's necessary to change the GMP sources if
this is a problem.
Language Bindings
*****************
The following packages and projects offer access to GMP from
languages other than C, though perhaps with varying levels of
functionality and efficiency.
C++
* GMP C++ class interface, *note C++ Class Interface::
Straightforward interface, expression templates to eliminate
temporaries.
* ALP `http://www.inria.fr/saga/logiciels/ALP'
Linear algebra and polynomials using templates.
* CLN `http://clisp.cons.org/~haible/packages-cln.html'
High level classes for arithmetic.
* LiDIA `http://www.informatik.tu-darmstadt.de/TI/LiDIA'
A C++ library for computational number theory.
* NTL `http://www.shoup.net/ntl'
A C++ number theory library.
Fortran
* Omni F77 `http://pdplab.trc.rwcp.or.jp/pdperf/Omni/home.html'
Arbitrary precision floats.
Haskell
* Glasgow Haskell Compiler `http://www.haskell.org/ghc'
Java
* Kaffe `http://www.kaffe.org'
* Kissme `http://kissme.sourceforge.net'
Lisp
* GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html'
In the process of switching to GMP for bignums.
* Librep `http://librep.sourceforge.net'
M4
* GNU m4 betas `http://www.seindal.dk/rene/gnu'
Optionally provides an arbitrary precision `mpeval'.
ML
* MLton compiler `http://www.sourcelight.com/MLton'
Oz
* Mozart `http://www.mozart-oz.org'
Pascal
* GNU Pascal Compiler `http://www.gnu-pascal.de'
GMP unit.
Perl
* GMP module, see `demos/perl' in the GMP sources.
* Math::GMP `http://www.cpan.org'
Compatible with Math::BigInt, but not as many functions as
the GMP module above.
* Math::BigInt::GMP `http://www.cpan.org'
Plug Math::GMP into normal Math::BigInt operations.
Pike
* mpz module in the standard distribution,
`http://pike.idonex.com'
Prolog
* SWI Prolog `http://www.swi.psy.uva.nl/projects/SWI-Prolog'
Arbitrary precision floats.
Python
* mpz module in the standard distribution,
`http://www.python.org'
* GMPY `http://gmpy.sourceforge.net'
Scheme
* RScheme `http://www.rscheme.org'
* STklos `http://kaolin.unice.fr/STklos'
Smalltalk
* GNU Smalltalk
`http://www.smalltalk.org/versions/GNUSmalltalk.html'
Other
* DrGenius `http://drgenius.seul.org'
Geometry system and mathematical programming language.
* GiNaC `http://www.ginac.de'
C++ computer algebra using CLN.
* Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
Macsyma computer algebra using GCL.
* Q `http://www.musikwissenschaft.uni-mainz.de/~ag/q'
Equational programming system.
* Regina `http://regina.sourceforge.net'
Topological calculator.
* Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
Yet another computer algebra system.
Algorithms
**********
This chapter is an introduction to some of the algorithms used for
various GMP operations. The code is likely to be hard to understand
without knowing something about the algorithms.
Some GMP internals are mentioned, but applications that expect to be
compatible with future GMP releases should take care to use only the
documented functions.
Multiplication
==============
NxN limb multiplications and squares are done using one of four
algorithms, as the size N increases.
Algorithm Threshold
Basecase (none)
Karatsuba `MUL_KARATSUBA_THRESHOLD'
Toom-3 `MUL_TOOM3_THRESHOLD'
FFT `MUL_FFT_THRESHOLD'
Similarly for squaring, with the `SQR' thresholds. Note though that
the FFT is only used if GMP is configured with `--enable-fft', *note
Build Options::.
NxM multiplications of operands with different sizes above
`MUL_KARATSUBA_THRESHOLD' are currently done by splitting into MxM
pieces. The Karatsuba and Toom-3 routines then operate only on equal
size operands. This is not very efficient, and is slated for
improvement in the future.
Basecase Multiplication
-----------------------
Basecase NxM multiplication is a straightforward rectangular set of
cross-products, the same as long multiplication done by hand and for
that reason sometimes known as the schoolbook or grammar school method.
This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M
(*note References::), and the `mpn/generic/mul_basecase.c' code.
Assembler implementations of `mpn_mul_basecase' are essentially the
same as the generic C code, but have all the usual assembler tricks and
obscurities introduced for speed.
A square can be done in roughly half the time of a multiply, by
using the fact that the cross products above and below the diagonal are
the same. A triangle of products below the diagonal is formed, doubled
(left shift by one bit), and then the products on the diagonal added.
This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembler
implementations take essentially the same approach.
u0 u1 u2 u3 u4
+---+---+---+---+---+
u0 | d | | | | |
+---+---+---+---+---+
u1 | | d | | | |
+---+---+---+---+---+
u2 | | | d | | |
+---+---+---+---+---+
u3 | | | | d | |
+---+---+---+---+---+
u4 | | | | | d |
+---+---+---+---+---+
In practice squaring isn't a full 2x faster than multiplying, it's
usually around 1.5x. Less than 1.5x probably indicates
`mpn_sqr_basecase' wants improving on that CPU.
On some CPUs `mpn_mul_basecase' can be faster than the generic C
`mpn_sqr_basecase'. `SQR_BASECASE_THRESHOLD' is the size at which to
use `mpn_sqr_basecase', this will be zero if that routine should be
used always.
Karatsuba Multiplication
------------------------
The Karatsuba multiplication algorithm is described in Knuth section
4.3.3 part A, and various other textbooks. A brief description is
given here.
The inputs x and y are treated as each split into two parts of equal
length (or the most significant part one limb shorter if N is odd).
high low
+----------+----------+
| x1 | x0 |
+----------+----------+
+----------+----------+
| y1 | y0 |
+----------+----------+
Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
y=y1*b+y0, and the following holds,
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
This formula means doing only three multiplies of (N/2)x(N/2) limbs,
whereas a basecase multiply of NxN limbs is equivalent to four
multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
positions where the three products must be added.
high low
+--------+--------+ +--------+--------+
| x1*y1 | | x0*y0 |
+--------+--------+ +--------+--------+
+--------+--------+
add | x1*y1 |
+--------+--------+
+--------+--------+
add | x0*y0 |
+--------+--------+
+--------+--------+
sub | (x1-x0)*(y1-y0) |
+--------+--------+
The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
and the sign used to choose to add or subtract. Notice the sum
high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
additions, rather than 6*k, but in GMP extra function call overheads
outweigh the saving.
Squaring is similar to multiplying, but with x=y the formula reduces
to an equivalent with three squares,
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
The final result is accumulated from those three squares the same
way as for the three multiplies above. The middle term (x1-x0)^2 is now
always positive.
A similar formula for both multiplying and squaring can be
constructed with a middle term (x1+x0)*(y1+y0). But those sums can
exceed k limbs, leading to more carry handling and additions than the
form above.
Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
the exponent being log(3)/log(2), representing 3 multiplies each 1/2
the size of the inputs. This is a big improvement over the basecase
multiply at O(N^2) and the advantage soon overcomes the extra additions
Karatsuba performs.
`MUL_KARATSUBA_THRESHOLD' can be as little as 10 limbs. The `SQR'
threshold is usually about twice the `MUL'. The basecase algorithm will
take a time of the form M(N) = a*N^2 + b*N + c and the Karatsuba
algorithm K(N) = 3*M(N/2) + d*N + e. Clearly per-crossproduct speedups
in the basecase code reduce a and decrease the threshold, but linear
style speedups reducing b will actually increase the threshold. The
latter can be seen for instance when adding an optimized
`mpn_sqr_diagonal' to `mpn_sqr_basecase'. Of course all speedups
reduce total time, and in that sense the algorithm thresholds are
merely of academic interest.
Toom-Cook 3-Way Multiplication
------------------------------
The Karatsuba formula is the simplest case of a general approach to
splitting inputs that leads to both Toom-Cook and FFT algorithms. A
description of Toom-Cook can be found in Knuth section 4.3.3, with an
example 3-way calculation after Theorem A. The 3-way form used in GMP
is described here.
The operands are each considered split into 3 pieces of equal length
(or the most significant part 1 or 2 limbs shorter than the others).
high low
+----------+----------+----------+
| x2 | x1 | x0 |
+----------+----------+----------+
+----------+----------+----------+
| y2 | y1 | y0 |
+----------+----------+----------+
These parts are treated as the coefficients of two polynomials
X(t) = x2*t^2 + x1*t + x0
Y(t) = y2*t^2 + y1*t + y0
Again let b equal the power of 2 which is the size of the x0, x1, y0
and y1 pieces, ie. if they're k limbs each then
b=2^(k*mp_bits_per_limb). With this x=X(b) and y=Y(b).
Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
The w[i] are going to be determined, and when they are they'll give the
final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The coefficients
will be roughly b^2 each, and the final W(b) will be an addition like,
high low
+-------+-------+
| w4 |
+-------+-------+
+--------+-------+
| w3 |
+--------+-------+
+--------+-------+
| w2 |
+--------+-------+
+--------+-------+
| w1 |
+--------+-------+
+-------+-------+
| w0 |
+-------+-------+
The w[i] coefficients could be formed by a simple set of cross
products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
equivalent merely to a basecase multiply. Instead the following
approach is used.
X(t) and Y(t) are evaluated and multiplied at 5 points, giving
values of W(t) at those points. The points used can be chosen in
various ways, but in GMP the following are used
Point Value
t=0 x0*y0, which gives w0 immediately
t=2 (4*x2+2*x1+x0)*(4*y2+2*y1+y0)
t=1 (x2+x1+x0)*(y2+y1+y0)
t=1/2 (x2+2*x1+4*x0)*(y2+2*y1+4*y0)
t=inf x2*y2, which gives w4 immediately
At t=1/2 the value calculated is actually 16*X(1/2)*Y(1/2), giving a
value for 16*W(1/2), and this is always an integer. At t=inf the value
is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
it's much easier to think of as simply x2*y2 giving w4 immediately
(much like x0*y0 at t=0 gives w0 immediately).
Now each of the points substituted into W(t)=w4*t^4+...+w0 gives a
linear combination of the w[i] coefficients, and the value of those
combinations has just been calculated.
W(0) = w0
16*W(1/2) = w4 + 2*w3 + 4*w2 + 8*w1 + 16*w0
W(1) = w4 + w3 + w2 + w1 + w0
W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
W(inf) = w4
This is a set of five equations in five unknowns, and some
elementary linear algebra quickly isolates each w[i], by subtracting
multiples of one equation from another.
In the code the set of five values W(0),...,W(inf) will represent
those certain linear combinations. By adding or subtracting one from
another as necessary, values which are each w[i] alone are arrived at.
This involves only a few subtractions of small multiples (some of which
are powers of 2), and so is fast. A couple of divisions remain by
powers of 2 and one division by 3 (or by 6 rather), and that last uses
the special `mpn_divexact_by3' (*note Exact Division::).
In the code the values w4, w2 and w0 are formed in the destination
with pointers `E', `C' and `A', and w3 and w1 in temporary space `D'
and `B' are added to them. There are extra limbs `tD', `tC' and `tB'
at the high end of w3, w2 and w1 which are handled separately. The
final addition then is as follows.
high low
+-------+-------+-------+-------+-------+-------+
| E | C | A |
+-------+-------+-------+-------+-------+-------+
+------+-------++------+-------+
| D || B |
+------+-------++------+-------+
-- -- --
|tD| |tC| |tB|
-- -- --
The conversion of W(t) values to the coefficients is interpolation.
A polynomial of degree 4 like W(t) is uniquely determined by values
known at 5 different points. The points can be chosen to make the
linear equations come out with a convenient set of steps for isolating
the w[i].
In `mpn/generic/mul_n.c' the `interpolate3' routine performs the
interpolation. The open-coded one-pass version may be a bit hard to
understand, the steps performed can be better seen in the `USE_MORE_MPN'
version.
Squaring follows the same procedure as multiplication, but there's
only one X(t) and it's evaluated at 5 points, and those values squared
to give values of W(t). The interpolation is then identical, and in
fact the same `interpolate3' subroutine is used for both squaring and
multiplying.
Toom-3 is asymptotically O(N^1.465), the exponent being
log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
size. This is an improvement over Karatsuba at O(N^1.585), though
Toom-Cook does more work in the evaluation and interpolation and so it
only realizes its advantage above a certain size.
Near the crossover between Toom-3 and Karatsuba there's generally a
range of sizes where the difference between the two is small.
`MUL_TOOM3_THRESHOLD' is a somewhat arbitrary point in that range and
successive runs of the tune program can give different values due to
small variations in measuring. A graph of time versus size for the two
shows the effect, see `tune/README'.
At the fairly small sizes where the Toom-3 thresholds occur it's
worth remembering that the asymptotic behaviour for Karatsuba and
Toom-3 can't be expected to make accurate predictions, due of course to
the big influence of all sorts of overheads, and the fact that only a
few recursions of each are being performed. Even at large sizes
there's a good chance machine dependent effects like cache architecture
will mean actual performance deviates from what might be predicted.
The formula given above for the Karatsuba algorithm has an
equivalent for Toom-3 involving only five multiplies, but this would be
complicated and unenlightening.
An alternate view of Toom-3 can be found in Zuras (*note
References::), using a vector to represent the x and y splits and a
matrix multiplication for the evaluation and interpolation stages. The
matrix inverses are not meant to be actually used, and they have
elements with values much greater than in fact arise in the
interpolation steps. The diagram shown for the 3-way is attractive,
but again doesn't have to be implemented that way and for example with
a bit of rearrangement just one division by 6 can be done.
FFT Multiplication
------------------
At large to very large sizes a Fermat style FFT multiplication is
used, following Scho"nhage and Strassen (*note References::).
Descriptions of FFTs in various forms can be found in many textbooks,
for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief
description of the form used in GMP is given here.
The multiplication done is x*y mod 2^N+1, for a given N. A full
product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x
and y with high zero limbs. The modular product is the native form for
the algorithm, so padding to get a full product is unavoidable.
The algorithm follows a split, evaluate, pointwise multiply,
interpolate and combine similar to that described above for Karatsuba
and Toom-3. A k parameter controls the split, with an FFT-k splitting
into 2^k pieces of M=N/2^k bits each. N must be a multiple of
(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
bit shifts in the split and combine stages.
The evaluations, pointwise multiplications, and interpolation, are
all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
2^k and of `mp_bits_per_limb'. The results of interpolation will be
the following negacyclic convolution of the input pieces, and the
choice of N' ensures these sums aren't truncated.
---
\ b
w[n] = / (-1) * x[i] * y[j]
---
i+j==b*2^k+n
b=0,1
The points used for the evaluation are g^i for i=0 to 2^k-1 where
g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
necessary cancellations at the interpolation stage, and it's also a
power of 2 so the fast fourier transforms used for the evaluation and
interpolation do only shifts, adds and negations.
The pointwise multiplications are done modulo 2^N'+1 and either
recurse into a further FFT or use a plain multiplication (Toom-3,
Karatsuba or basecase), whichever is optimal at the size N'. The
interpolation is an inverse fast fourier transform. The resulting set
of sums of x[i]*y[j] are added at appropriate offsets to give the final
result.
Squaring is the same, but x is the only input so it's one transform
at the evaluate stage and the pointwise multiplies are squares. The
interpolation is the same.
For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm,
the exponent representing 2^k recursed modular multiplies each
1/2^(k-1) the size of the original. Each successive k is an asymptotic
improvement, but overheads mean each is only faster at bigger and
bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the
thresholds where each k is used. Each new k effectively swaps some
multiplying for some shifts, adds and overheads.
A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
A k=4 FFT at O(N^1.333) can be expected to be the first faster than
Toom-3 at O(N^1.465). In practice this is what's found, with
`MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300
and 1000 limbs, depending on the CPU. So far it's been found that only
very large FFTs recurse into pointwise multiplies above these sizes.
When an FFT is to give a full product, the change of N to 2N doesn't
alter the theoretical complexity for a given k, but for the purposes of
considering where an FFT might be first used it can be assumed that the
FFT is recursing into a normal multiply and that on that basis it's
doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
first FFT faster than Toom-3. In practice `MUL_FFT_THRESHOLD' and
`SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere
between 3000 and 10000 limbs.
The way N is split into 2^k pieces and then 2M+k+3 is rounded up to
a multiple of 2^k and `mp_bits_per_limb' means that when
2^k>=mp_bits_per_limb the effective N is a multiple of 2^(2k-1) bits.
The +k+3 means some values of N just under such a multiple will be
rounded to the next. The complexity calculations above assume that a
favourable size is used, meaning one which isn't padded through
rounding, and it's also assumed that the extra +k+3 bits are negligible
at typical FFT sizes.
The practical effect of the 2^(2k-1) constraint is to introduce a
step-effect into measured speeds. For example k=8 will round N up to a
multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
groups of sizes for which `mpn_mul_n' runs at the same speed. Or for
k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice
it's been found each k is used at quite small multiples of its size
constraint and so the step effect is quite noticeable in a time versus
size graph.
The threshold determinations currently measure at the mid-points of
size steps, but this is sub-optimal since at the start of a new step it
can happen that it's better to go back to the previous k for a while.
Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE'
will be needed.
Other Multiplication
--------------------
The 3-way Toom-Cook algorithm described above (*note Toom-Cook 3-Way
Multiplication::) generalizes to split into an arbitrary number of
pieces, as per Knuth section 4.3.3 algorithm C. This is not currently
used, though it's possible a Toom-4 might fit in between Toom-3 and the
FFTs. The notes here are merely for interest.
In general a split into r+1 pieces is made, and evaluations and
pointwise multiplications done at 2*r+1 points. A 4-way split does 7
pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
multiplications count towards big-O complexity, but the time spent in
the evaluate and interpolate stages grows with r and has a significant
practical impact, with the asymptotic advantage of each r realized only
at bigger and bigger sizes. The overheads grow as O(N*r), whereas in
an r=2^k FFT they grow only as O(N*log(r)).
Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
uses -r,...,0,...,r and the latter saves some small multiplies in the
evaluate stage (or rather trades them for additions), and has a further
saving of nearly half the interpolate steps. The idea is to separate
odd and even final coefficients and then perform algorithm C steps C7
and C8 on them separately. The divisors at step C7 become j^2 and the
multipliers at C8 become 2*t*j-j^2.
Splitting odd and even parts through positive and negative points
can be thought of as using -1 as a square root of unity. If a 4th root
of unity was available then a further split and speedup would be
possible, but no such root exists for plain integers. Going to complex
integers with i=sqrt(-1) doesn't help, essentially because in cartesian
form it takes three real multiplies to do a complex multiply. The
existence of 2^k'th roots of unity in a suitable ring or field lets the
fast fourier transform keep splitting and get to O(N*log(r)).
Floating point FFTs use complex numbers approximating Nth roots of
unity. Some processors have special support for such FFTs. But these
are not used in GMP since it's very difficult to guarantee an exact
result (to some number of bits). An occasional difference of 1 in the
last bit might not matter to a typical signal processing algorithm, but
is of course of vital importance to GMP.
Division Algorithms
===================
Single Limb Division
--------------------
Nx1 division is implemented using repeated 2x1 divisions from high
to low, either with a hardware divide instruction or a multiplication by
inverse, whichever is best on a given CPU.
The multiply by inverse follows section 8 of "Division by Invariant
Integers using Multiplication" by Granlund and Montgomery (*note
References::) and is implemented as `udiv_qrnnd_preinv' in
`gmp-impl.h'. The idea is to have a fixed-point approximation to 1/d
(see `invert_limb') and then multiply by the high limb (plus one bit)
of the dividend to get a quotient q. With d normalized (high bit set),
q is no more than 1 too small. Subtracting q*d from the dividend gives
a remainder, and reveals whether q or q-1 is correct.
The result is a division done with two multiplications and four or
five arithmetic operations. On CPUs with low latency multipliers this
can be much faster than a hardware divide, though the cost of
calculating the inverse at the start may mean it's only better on
inputs bigger than say 4 or 5 limbs.
When a divisor must be normalized, either for the generic C
`__udiv_qrnnd_c' or the multiply by inverse, the division performed is
actually a*2^k by d*2^k where a is the dividend and k is the power
necessary to have the high bit of d*2^k set. The bit shifts for the
dividend are usually accomplished "on the fly" meaning by extracting
the appropriate bits at each step. Done this way the quotient limbs
come out aligned ready to store. When only the remainder is wanted, an
alternative is to take the dividend limbs unshifted and calculate r = a
mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
help on CPUs with poor bit shifts or few registers.
The multiply by inverse can be done two limbs at a time. The
calculation is basically the same, but the inverse is two limbs and the
divisor treated as if padded with a low zero limb. This means more
work, since the inverse will need a 2x2 multiply, but the four 1x1s to
do that are independent and can therefore be done partly or wholly in
parallel. Likewise for a 2x1 calculating q*d. The net effect is to
process two limbs with roughly the same two multiplies worth of latency
that one limb at a time gives. This extends to 3 or 4 limbs at a time,
though the extra work to apply the inverse will almost certainly soon
reach the limits of multiplier throughput.
A similar approach in reverse can be taken to process just half a
limb at a time if the divisor is only a half limb. In this case the
1x1 multiply for the inverse effectively becomes two (1/2)x1 for each
limb, which can be a saving on CPUs with a fast half limb multiply, or
in fact if the only multiply is a half limb, and especially if it's not
pipelined.
Basecase Division
-----------------
Basecase NxM division is like long division done by hand, but in base
2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and
`mpn/generic/sb_divrem_mn.c'.
Briefly stated, while the dividend remains larger than the divisor,
a high quotient limb is formed and the Nx1 product q*d subtracted at
the top end of the dividend. With a normalized divisor (most
significant bit set), each quotient limb can be formed with a 2x1
division and a 1x1 multiplication plus some subtractions. The 2x1
division is by the high limb of the divisor and is done either with a
hardware divide or a multiply by inverse (the same as in *Note Single
Limb Division::) whichever is faster. Such a quotient is sometimes one
too big, requiring an addback of the divisor, but that happens rarely.
With Q=N-M being the number of quotient limbs, this is an O(Q*M)
algorithm and will run at a speed similar to a basecase QxM
multiplication, differing in fact only in the extra multiply and divide
for each of the Q quotient limbs.
Divide and Conquer Division
---------------------------
For divisors larger than `DIV_DC_THRESHOLD', division is done by
dividing. Or to be precise by a recursive divide and conquer algorithm
based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
(*note References::).
The algorithm consists essentially of recognising that a 2NxN
division can be done with the basecase division algorithm (*note
Basecase Division::), but using N/2 limbs as a base, not just a single
limb. This way the multiplications that arise are (N/2)x(N/2) and can
take advantage of Karatsuba and higher multiplication algorithms (*note
Multiplication Algorithms::). The "digits" of the quotient are formed
by recursive Nx(N/2) divisions.
If the (N/2)x(N/2) multiplies are done with a basecase multiplication
then the work is about the same as a basecase division, but with more
function call overheads and with some subtractions separated from the
multiplies. These overheads mean that it's only when N/2 is above
`MUL_KARATSUBA_THRESHOLD' that divide and conquer is of use.
`DIV_DC_THRESHOLD' is based on the divisor size N, so it will be
somewhere above twice `MUL_KARATSUBA_THRESHOLD', but how much above
depends on the CPU. An optimized `mpn_mul_basecase' can lower
`DIV_DC_THRESHOLD' a little by offering a ready-made advantage over
repeated `mpn_submul_1' calls.
Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
the time for an NxN multiplication done with FFTs. The actual time is
a sum over multiplications of the recursed sizes, as can be seen near
the end of section 2.2 of Burnikel and Ziegler. For example, within
the Toom-3 range, divide and conquer is 2.63*M(N). With higher
algorithms the M(N) term improves and the multiplier tends to log(N).
In practice, at moderate to large sizes, a 2NxN division is about 2 to
4 times slower than an NxN multiplication.
Newton's method used for division is asymptotically O(M(N)) and
should therefore be superior to divide and conquer, but it's believed
this would only be for large to very large N.
Exact Division
--------------
A so-called exact division is when the dividend is known to be an
exact multiple of the divisor. Jebelean's exact division algorithm
uses this knowledge to make some significant optimizations (*note
References::).
The idea can be illustrated in decimal for example with 368154
divided by 543. Because the low digit of the dividend is 4, the low
digit of the quotient must be 8. This is arrived at from 4*7 mod 10,
using the fact 7 is the modular inverse of 3 (the low digit of the
divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from
the dividend leaving 363810. Notice the low digit has become zero.
The procedure is repeated at the second digit, with the next
quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving
325800. And finally at the third digit with quotient digit 6 (8*7 mod
10), subtracting 6*543=3258 leaving 0. So the quotient is 678.
Notice however that the multiplies and subtractions don't need to
extend past the low three digits of the dividend, since that's enough
to determine the three quotient digits. For the last quotient digit no
subtraction is needed at all. On a 2NxN division like this one, only
about half the work of a normal basecase division is necessary.
For an NxM exact division producing Q=N-M quotient limbs, the saving
over a normal basecase division is in two parts. Firstly, each of the
Q quotient limbs needs only one multiply, not a 2x1 divide and
multiply. Secondly, the crossproducts are reduced when Q>M to
Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are
complementary. If Q is big then many divisions are saved, or if Q is
small then the crossproducts reduce to a small number.
The modular inverse used is calculated efficiently by
`modlimb_invert' in `gmp-impl.h'. This does four multiplies for a
32-bit limb, or six for a 64-bit limb. `tune/modlinv.c' has some
alternate implementations that might suit processors better at bit
twiddling than multiplying.
The sub-quadratic exact division described by Jebelean in "Exact
Division with Karatsuba Complexity" is not currently implemented. It
uses a rearrangement similar to the divide and conquer for normal
division (*note Divide and Conquer Division::), but operating from low
to high. A further possibility not currently implemented is
"Bidirectional Exact Integer Division" by Krandick and Jebelean which
forms quotient limbs from both the high and low ends of the dividend,
and can halve once more the number of crossproducts needed in a 2NxN
division.
A special case exact division by 3 exists in `mpn_divexact_by3',
supporting Toom-3 multiplication and `mpq' canonicalizations. It forms
quotient digits with a multiply by the modular inverse of 3 (which is
`0xAA..AAB') and uses two comparisons to determine a borrow for the next
limb. The multiplications don't need to be on the dependent chain, as
long as the effect of the borrows is applied. Only a few optimized
assembler implementations currently exist.
Exact Remainder
---------------
If the exact division algorithm is done with a full subtraction at
each stage and the dividend isn't a multiple of the divisor, then low
zero limbs are produced but with a remainder in the high limbs. For
dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, then this
remainder r is of the form
a = q*d + r*b^n
n represents the number of zero limbs produced by the subtractions,
that being the number of limbs produced for q. r will be in the range
0<=rb*r+u2 condition appropriately relaxed.
Greatest Common Divisor
=======================
Binary GCD
----------
At small sizes GMP uses an O(N^2) binary style GCD. This is
described in many textbooks, for example Knuth section 4.5.2 algorithm
B. It simply consists of successively reducing operands a and b using
gcd(a,b) = gcd(min(a,b),abs(a-b)), and also that if a and b are first
made odd then abs(a-b) is even and factors of two can be discarded.
Variants like letting a-b become negative and doing a different next
step are of interest only as far as they suit particular CPUs, since on
small operands it's machine dependent factors that determine
performance.
The Euclidean GCD algorithm, as per Knuth algorithms E and A,
reduces using a mod b but this has so far been found to be slower
everywhere. One reason the binary method does well is that the implied
quotient at each step is usually small, so often only one or two
subtractions are needed to get the same effect as a division.
Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth
section 4.5.3 Theorem E.
When the implied quotient is large, meaning b is much smaller than
a, then a division is worthwhile. This is the basis for the initial a
mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
and 1x1 cases). But after that initial reduction, big quotients occur
too rarely to make it worth checking for them.
Accelerated GCD
---------------
For sizes above `GCD_ACCEL_THRESHOLD', GMP uses the Accelerated GCD
algorithm described independently by Weber and Jebelean (the latter as
the "Generalized Binary" algorithm), *note References::. This
algorithm is still O(N^2), but is much faster than the binary algorithm
since it does fewer multi-precision operations. It consists of
alternating the k-ary reduction by Sorenson, and a "dmod" exact
remainder reduction.
For operands u and v the k-ary reduction replaces u with n*v-d*u
where n and d are single limb values chosen to give two trailing zero
limbs on that value, which can be stripped. n and d are calculated
using an algorithm similar to half of a two limb GCD (see `find_a' in
`mpn/generic/gcd.c').
When u and v differ in size by more than a certain number of bits, a
dmod is performed to zero out bits at the low end of the larger. It
consists of an exact remainder style division applied to an appropriate
number of bits (*note Exact Division::, and *note Exact Remainder::).
This is faster than a k-ary reduction but useful only when the operands
differ in size. There's a dmod after each k-ary reduction, and if the
dmod leaves the operands still differing in size then it's repeated.
The k-ary reduction step can introduce spurious factors into the GCD
calculated, and these are eliminated at the end by taking GCDs with the
original inputs gcd(u,gcd(v,g)) using the binary algorithm. Since g is
almost always small this takes very little time.
At small sizes the algorithm needs a good implementation of
`find_a'. At larger sizes it's dominated by `mpn_addmul_1' applying n
and d.
Extended GCD
------------
The extended GCD calculates gcd(a,b) and also cofactors x and y
satisfying a*x+b*y=gcd(a,b). Lehmer's multi-step improvement of the
extended Euclidean algorithm is used. See Knuth section 4.5.2
algorithm L, and `mpn/generic/gcdext.c'. This is an O(N^2) algorithm.
The multipliers at each step are found using single limb
calculations for sizes up to `GCDEXT_THRESHOLD', or double limb
calculations above that. The single limb code is faster but doesn't
produce full-limb multipliers, hence not making full use of the
`mpn_addmul_1' calls.
When a CPU has a data-dependent multiplier, meaning one which is
faster on operands with fewer bits, the extra work in the double-limb
calculation might only save some looping overheads, leading to a large
`GCDEXT_THRESHOLD'.
Currently the single limb calculation doesn't optimize for the small
quotients that often occur, and this can lead to unusually low values of
`GCDEXT_THRESHOLD', depending on the CPU.
An analysis of double-limb calculations can be found in "A
Double-Digit Lehmer-Euclid Algorithm" by Jebelean (*note References::).
The code in GMP was developed independently.
It should be noted that when a double limb calculation is used, it's
used for the whole of that GCD, it doesn't fall back to single limb
part way through. This is because as the algorithm proceeds, the
inputs a and b are reduced, but the cofactors x and y grow, so the
multipliers at each step are applied to a roughly constant total number
of limbs.
Jacobi Symbol
-------------
`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
simple binary algorithm similar to that described for the GCDs (*note
Binary GCD::). They're not very fast when both inputs are large.
Lehmer's multi-step improvement or a binary based multi-step algorithm
is likely to be better.
When one operand fits a single limb, and that includes
`mpz_kronecker_ui' and friends, an initial reduction is done with
either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
algorithm on a single limb. The binary algorithm is well suited to a
single limb, and the whole calculation in this case is quite efficient.
In all the routines sign changes for the result are accumulated
using some bit twiddling, avoiding table lookups or conditional jumps.
Powering Algorithms
===================
Normal Powering
---------------
Normal `mpz' or `mpf' powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is
seen in the exponent, as per Knuth section 4.6.3. The "left to right"
variant described there is used rather than algorithm A, since it's
just as easy and can be done with somewhat less temporary memory.
Modular Powering
----------------
Modular powering is implemented using a 2^k-ary sliding window
algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
(*note References::). k is chosen according to the size of the
exponent. Larger exponents use larger values of k, the choice being
made to minimize the average number of multiplications that must
supplement the squaring.
The modular multiplies and squares use either a simple division or
the REDC method by Montgomery (*note References::). REDC is a little
faster, essentially saving N single limb divisions in a fashion similar
to an exact remainder (*note Exact Remainder::). The current REDC has
some limitations. It's only O(N^2) so above `POWM_THRESHOLD' division
becomes faster and is used. It doesn't attempt to detect small bases,
but rather always uses a REDC form, which is usually a full size
operand. And lastly it's only applied to odd moduli.
Root Extraction Algorithms
==========================
Square Root
-----------
Square roots are taken using the "Karatsuba Square Root" algorithm
by Paul Zimmermann (*note References::). This is expressed in a divide
and conquer form, but as noted in the paper it can also be viewed as a
discrete variant of Newton's method.
In the Karatsuba multiplication range this is an O(1.5*M(N/2))
algorithm, where M(n) is the time to multiply two numbers of n limbs.
In the FFT multiplication range this grows to a bound of O(6*M(N/2)).
In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and
Toom-3 ranges, growing to 2 or 3 in the FFT range.
The algorithm does all its calculations in integers and the resulting
`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
Nth Root
--------
Integer Nth roots are taken using Newton's method with the following
iteration, where A is the input and n is the root to be taken.
1 A
a[i+1] = - * ( --------- + (n-1)*a[i] )
n a[i]^(n-1)
The initial approximation a[1] is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just
above the true root. The iteration converges quadratically when
started from a good approximation. When n is large more initial bits
are needed to get good convergence. The current implementation is not
particularly well optimized.
Perfect Square
--------------
`mpz_perfect_square_p' is able to quickly exclude most non-squares by
checking whether the input is a quadratic residue modulo some small
integers.
The first test is modulo 256 which means simply examining the least
significant byte. Only 44 different values occur as the low byte of a
square, so 82.8% of non-squares can be immediately excluded. Similar
tests modulo primes from 3 to 29 exclude 99.5% of those remaining, or
if a limb is 64 bits then primes up to 53 are used, excluding 99.99%.
A single Nx1 remainder using `PP' from `gmp-impl.h' quickly gives all
these remainders.
A square root must still be taken for any value that passes the
residue tests, to verify it's really a square and not one of the 0.086%
(or 0.000156% for 64 bits) non-squares that get through. *Note Square
Root Algorithm::.
Perfect Power
-------------
Detecting perfect powers is required by some factorization
algorithms. Currently `mpz_perfect_power_p' is implemented using
repeated Nth root extractions, though naturally only prime roots need
to be considered. (*Note Nth Root Algorithm::.)
If a prime divisor p with multiplicity e can be found, then only
roots which are divisors of e need to be considered, much reducing the
work necessary. To this end divisibility by a set of small primes is
checked.
Radix Conversion
================
Radix conversions are less important than other algorithms. A
program dominated by conversions should probably use a different data
representation.
Binary to Radix
---------------
Conversions from binary to a power-of-2 radix use a simple and fast
O(N) bit extraction algorithm.
Conversions from binary to other radices use one of two algorithms.
Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
Repeated divisions by b^n are made, where b is the radix and n is the
biggest power that fits in a limb. But instead of simply using the
remainder r from such divisions, an extra divide step is done to give a
fractional limb representing r/b^n. The digits of r can then be
extracted using multiplications by b rather than divisions. Special
case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.
Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used. For an input t, powers b^(n*2^i) of the radix are calculated,
until a power between t and sqrt(t) is reached. t is then divided by
that largest power, giving a quotient which is the digits above that
power, and a remainder which is those below. These two parts are in
turn divided by the second highest power, and so on recursively. When
a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
limbs, the basecase algorithm described above is used.
The advantage of this algorithm is that big divisions can make use
of the sub-quadratic divide and conquer division (*note Divide and
Conquer Division::), and big divisions tend to have less overheads than
lots of separate single limb divisions anyway. But in any case the
cost of calculating the powers b^(n*2^i) must first be overcome.
`GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
the same basic thing, the point where it becomes worth doing a big
division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
includes the cost of calculating the radix power required, whereas
`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
case when recursing.
Since the base case produces digits from least to most significant
but they want to be stored from most to least, it's necessary to
calculate in advance how many digits there will be, or at least be sure
not to underestimate that. For GMP the number of input bits is
multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
The result is either correct or one too big.
Examining some of the high bits of the input could increase the
chance of getting the exact number of digits, but an exact result every
time would not be practical, since in general the difference between
numbers 100... and 99... is only in the last few bits and the work to
identify 99... might well be almost as much as a full conversion.
`mpf_get_str' doesn't currently use the algorithm described here, it
multiplies or divides by a power of b to move the radix point to the
just above the highest non-zero digit (or at worst one above that
location), then multiplies by b^n to bring out digits. This is O(N^2)
and is certainly not optimal.
The r/b^n scheme described above for using multiplications to bring
out digits might be useful for more than a single limb. Some brief
experiments with it on the base case when recursing didn't give a
noticable improvement, but perhaps that was only due to the
implementation. Something similar would work for the sub-quadratic
divisions too, though there would be the cost of calculating a bigger
radix power.
Another possible improvement for the sub-quadratic part would be to
arrange for radix powers that balanced the sizes of quotient and
remainder produced, ie. the highest power would be an b^(n*k)
approximately equal to sqrt(t), not restricted to a 2^i factor. That
ought to smooth out a graph of times against sizes, but may or may not
be a net speedup.
Radix to Binary
---------------
Conversions from a power-of-2 radix into binary use a simple and fast
O(N) bitwise concatenation algorithm.
Conversions from other radices use one of two algorithms. Sizes
below `SET_STR_THRESHOLD' use a basic O(N^2) method. Groups of n
digits are converted to limbs, where n is the biggest power of the base
b which will fit in a limb, then those groups are accumulated into the
result by multiplying by b^n and adding. This saves multi-precision
operations, as per Knuth section 4.4 part E (*note References::). Some
special case code is provided for decimal, giving the compiler a chance
to optimize multiplications by 10.
Above `SET_STR_THRESHOLD' a sub-quadratic algorithm is used. First
groups of n digits are converted into limbs. Then adjacent limbs are
combined into limb pairs with x*b^n+y, where x and y are the limbs.
Adjacent limb pairs are combined into quads similarly with x*b^(2n)+y.
This continues until a single block remains, that being the result.
The advantage of this method is that the multiplications for each x
are big blocks, allowing Karatsuba and higher algorithms to be used.
But the cost of calculating the powers b^(n*2^i) must be overcome.
`SET_STR_THRESHOLD' usually ends up quite big, around 5000 digits, and
on some processors much bigger still.
`SET_STR_THRESHOLD' is based on the input digits (and tuned for
decimal), though it might be better based on a limb count, so as to be
independent of the base. But that sort of count isn't used by the base
case and so would need some sort of initial calculation or estimate.
The main reason `SET_STR_THRESHOLD' is so much bigger than the
corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that `mpn_mul_1' is
much faster than `mpn_divrem_1' (often by a factor of 10, or more).
Other Algorithms
================
Factorial
---------
Factorials n! are calculated by a simple product from 1 to n, but
arranged into certain sub-products.
First as many factors as fit in a limb are accumulated, then two of
those multiplied to give a 2-limb product. When two 2-limb products
are ready they're multiplied to a 4-limb product, and when two 4-limbs
are ready they're multiplied to an 8-limb product, etc. A stack of
outstanding products is built up, with two of the same size multiplied
together when ready.
Arranging for multiplications to have operands the same (or nearly
the same) size means the Karatsuba and higher multiplication algorithms
can be used. And even on sizes below the Karatsuba threshold an NxN
multiply will give a basecase multiply more to work on.
An obvious improvement not currently implemented would be to strip
factors of 2 from the products and apply them at the end with a bit
shift. Another possibility would be to determine the prime
factorization of the result (which can be done easily), and use a
powering method, at each stage squaring then multiplying in those
primes with a 1 in their exponent at that point. The advantage would
be some multiplies turned into squares.
Binomial Coefficients
---------------------
Binomial coefficients C(n,k) are calculated by first arranging k <=
n/2 using C(n,k) = C(n,n-k) if necessary, and then evaluating the
following product simply from i=2 to i=k.
k (n-k+i)
C(n,k) = (n-k+1) * prod -------
i=2 i
It's easy to show that each denominator i will divide the product so
far, so the exact division algorithm is used (*note Exact Division::).
The numerators n-k+i and denominators i are first accumulated into
as many fit a limb, to save multi-precision operations, though for
`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
and n-k+i in general won't fit in a limb at all.
An obvious improvement would be to strip factors of 2 from each
multiplier and divisor and count them separately, to be applied with a
bit shift at the end. Factors of 3 and perhaps 5 could even be handled
similarly. Another possibility, if n is not too big, would be to
determine the prime factorization of the result based on the factorials
involved, and power up those primes appropriately. This would help
most when k is near n/2.
Fibonacci Numbers
-----------------
The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed
for calculating isolated F[n] or F[n],F[n-1] values efficiently.
For small n, a table of single limb values in `__gmp_fib_table' is
used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
to F[93]. For convenience the table starts at F[-1].
Beyond the table, values are generated with a binary powering
algorithm, calculating a pair F[n] and F[n-1] working from high to low
across the bits of n. The formulas used are
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
F[2k-1] = F[k]^2 + F[k-1]^2
F[2k] = F[2k+1] - F[2k-1]
At each step, k is the high b bits of n. If the next bit of n is 0
then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
and the process repeated until all bits of n are incorporated. Notice
these formulas require just two squares per bit of n.
It'd be possible to handle the first few n above the single limb
table with simple additions, using the defining Fibonacci recurrence
F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
be faster for only about 10 or 20 values of n, and including a block of
code for just those doesn't seem worthwhile. If they really mattered
it'd be better to extend the data table.
Using a table avoids lots of calculations on small numbers, and
makes small n go fast. A bigger table would make more small n go fast,
it's just a question of balancing size against desired speed. For GMP
the code is kept compact, with the emphasis primarily on a good
powering algorithm.
`mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
interested in F[n]. In this case the last step of the algorithm can
become one multiply instead of two squares. One of the following two
formulas is used, according as n is odd or even.
F[2k] = F[k]*(F[k]+2F[k-1])
F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
F[2k+1] here is the same as above, just rearranged to be a multiply.
For interest, the 2*(-1)^k term both here and above can be applied
just to the low limb of the calculation, without a carry or borrow into
further limbs, which saves some code size. See comments with
`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
Lucas Numbers
-------------
`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
Fibonacci numbers with the following simple formulas.
L[k] = F[k] + 2*F[k-1]
L[k-1] = 2*F[k] - F[k-1]
`mpz_lucnum_ui' is only interested in L[n], and some work can be
saved. Trailing zero bits on n can be handled with a single square
each.
L[2k] = L[k]^2 - 2*(-1)^k
And the lowest 1 bit can be handled with one multiply of a pair of
Fibonacci numbers, similar to what `mpz_fib_ui' does.
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
Assembler Coding
================
The assembler subroutines in GMP are the most significant source of
speed at small to moderate sizes. At larger sizes algorithm selection
becomes more important, but of course speedups in low level routines
will still speed up everything proportionally.
Carry handling and widening multiplies that are important for GMP
can't be easily expressed in C. GCC `asm' blocks help a lot and are
provided in `longlong.h', but hand coding low level routines invariably
offers a speedup over generic C by a factor of anything from 2 to 10.
Code Organisation
-----------------
The various `mpn' subdirectories contain machine-dependent code,
written in C or assembler. The `mpn/generic' subdirectory contains
default code, used when there's no machine-specific version of a
particular file.
Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
64-bit variants in a family cannot share code and will have separate
directories. Within a family further subdirectories may exist for CPU
variants.
Assembler Basics
----------------
`mpn_addmul_1' and `mpn_submul_1' are the most important routines
for overall GMP performance. All multiplications and divisions come
down to repeated calls to these. `mpn_add_n', `mpn_sub_n',
`mpn_lshift' and `mpn_rshift' are next most important.
On some CPUs assembler versions of the internal functions
`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
mainly through avoiding function call overheads. They can also
potentially make better use of a wide superscalar processor.
The restrictions on overlaps between sources and destinations (*note
Low-level Functions::) are designed to facilitate a variety of
implementations. For example, knowing `mpn_add_n' won't have partly
overlapping sources and destination means reading can be done far ahead
of writing on superscalar processors, and loops can be vectorized on a
vector processor, depending on the carry handling.
Carry Propagation
-----------------
The problem that presents most challenges in GMP is propagating
carries from one limb to the next. In functions like `mpn_addmul_1' and
`mpn_add_n', carries are the only dependencies between limb operations.
On processors with carry flags, a straightforward CISC style `adc' is
generally best. AMD K6 `mpn_addmul_1' however is an example of an
unusual set of circumstances where a branch works out better.
On RISC processors generally an add and compare for overflow is
used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
carry propagation schemes require 4 instructions, meaning at least 4
cycles per limb, but other schemes may use just 1 or 2. On wide
superscalar processors performance may be completely determined by the
number of dependent instructions between carry-in and carry-out for
each limb.
On vector processors good use can be made of the fact that a carry
bit only very rarely propagates more than one limb. When adding a
single bit to a limb, there's only a carry out if that limb was
`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
`mpn/cray/add_n.c' is an example of this, it adds all limbs in
parallel, adds one set of carry bits in parallel and then only rarely
needs to fall through to a loop propagating further carries.
On the x86s, GCC (as of version 2.95.2) doesn't generate
particularly good code for the RISC style idioms that are necessary to
handle carry bits in C. Often conditional jumps are generated where
`adc' or `sbb' forms would be better. And so unfortunately almost any
loop involving carry bits needs to be coded in assembler for best
results.
Cache Handling
--------------
GMP aims to perform well both on operands that fit entirely in L1
cache and those which don't.
Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
large operands, so L2 and main memory performance is important for them.
`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
basecases, so L1 performance matters most for them, unless assembler
versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
case the remaining uses are mostly for larger operands.
For L2 or main memory operands, memory access times will almost
certainly be more than the calculation time. The aim therefore is to
maximize memory throughput, by starting a load of the next cache line
which processing the contents of the previous one. Clearly this is
only possible if the chip has a lock-up free cache or some sort of
prefetch instruction. Most current chips have both these features.
Prefetching sources combines well with loop unrolling, since a
prefetch can be initiated once per unrolled loop (or more than once if
the loop covers more than one cache line).
On CPUs without write-allocate caches, prefetching destinations will
ensure individual stores don't go further down the cache hierarchy,
limiting bandwidth. Of course for calculations which are slow anyway,
like `mpn_divrem_1', write-throughs might be fine.
The distance ahead to prefetch will be determined by memory latency
versus throughput. The aim of course is to have data arriving
continuously, at peak throughput. Some CPUs have limits on the number
of fetches or prefetches in progress.
If a special prefetch instruction doesn't exist then a plain load
can be used, but in that case care must be taken not to attempt to read
past the end of an operand, since that might produce a segmentation
violation.
Some CPUs or systems have hardware that detects sequential memory
accesses and initiates suitable cache movements automatically, making
life easy.
Floating Point
--------------
Floating point arithmetic is used in GMP for multiplications on CPUs
with poor integer multipliers. It's mostly useful for `mpn_mul_1',
`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
`mpn_mul_basecase' on both 32-bit and 64-bit machines.
With IEEE 53-bit double precision floats, integer multiplications
producing up to 53 bits will give exact results. Breaking a 64x64
multiplication into eight 16x32->48 bit pieces is convenient. With
some care though six 21x32->53 bit products can be used, if one of the
lower two 21-bit pieces also uses the sign bit.
For the `mpn_mul_1' family of functions on a 64-bit machine, the
invariant single limb is split at the start, into 3 or 4 pieces.
Inside the loop, the bignum operand is split into 32-bit pieces. Fast
conversion of these unsigned 32-bit pieces to floating point is highly
machine-dependent. In some cases, reading the data into the integer
unit, zero-extending to 64-bits, then transferring to the floating
point unit back via memory is the only option.
Converting partial products back to 64-bit limbs is usually best
done as a signed conversion. Since all values are smaller than 2^53,
signed and unsigned are the same, but most processors lack unsigned
conversions.
Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
into four 16-bit parts. The multi-limb operand U is split in the loop
into two 32-bit parts.
+---+---+---+---+
|v48|v32|v16|v00| V operand
+---+---+---+---+
+-------+---+---+
x | u32 | u00 | U operand (one limb)
+---------------+
---------------------------------
+-----------+
| u00 x v00 | p00 48-bit products
+-----------+
+-----------+
| u00 x v16 | p16
+-----------+
+-----------+
| u00 x v32 | p32
+-----------+
+-----------+
| u00 x v48 | p48
+-----------+
+-----------+
| u32 x v00 | r32
+-----------+
+-----------+
| u32 x v16 | r48
+-----------+
+-----------+
| u32 x v32 | r64
+-----------+
+-----------+
| u32 x v48 | r80
+-----------+
p32 and r32 can be summed using floating-point addition, and
likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from
the previous iteration.
For each loop then, four 49-bit quantities are transfered to the
integer unit, aligned as follows,
|-----64bits----|-----64bits----|
+------------+
| p00 + r64' | i00
+------------+
+------------+
| p16 + r80' | i16
+------------+
+------------+
| p32 + r32 | i32
+------------+
+------------+
| p48 + r48 | i48
+------------+
The challenge then is to sum these efficiently and add in a carry
limb, generating a low 64-bit result limb and a high 33-bit carry limb
(i48 extends 33 bits into the high half).
SIMD Instructions
-----------------
The single-instruction multiple-data support in current
microprocessors is aimed at signal processing algorithms where each
data point can be treated more or less independently. There's
generally not much support for propagating the sort of carries that
arise in GMP.
SIMD multiplications of say four 16x16 bit multiplies only do as much
work as one 32x32 from GMP's point of view, and need some shifts and
adds besides. But of course if say the SIMD form is fully pipelined
and uses less instruction decoding then it may still be worthwhile.
On the 80x86 chips, MMX has so far found a use in `mpn_rshift' and
`mpn_lshift' since it allows 64-bit operations, and is used in a special
case for 16-bit multipliers in the P55 `mpn_mul_1'. 3DNow and SSE
haven't found a use so far.
Software Pipelining
-------------------
Software pipelining consists of scheduling instructions around the
branch point in a loop. For example a loop taking a checksum of an
array of limbs might have a load and an add, but the load wouldn't be
for that add, rather for the one next time around the loop. Each load
then is effectively scheduled back in the previous iteration, allowing
latency to be hidden.
Naturally this is wanted only when doing things like loads or
multiplies that take a few cycles to complete, and only where a CPU has
multiple functional units so that other work can be done while waiting.
A pipeline with several stages will have a data value in progress at
each stage and each loop iteration moves them along one stage. This is
like juggling.
Within the loop some moves between registers may be necessary to
have the right values in the right places for each iteration. Loop
unrolling can help this, with each unrolled block able to use different
registers for different values, even if some shuffling is still needed
just before going back to the top of the loop.
Loop Unrolling
--------------
Loop unrolling consists of replicating code so that several limbs are
processed in each loop. At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for
example alternately using one register combination and then another.
Judicious use of `m4' macros can help avoid lots of duplication in the
source code.
Unrolling is commonly done to a power of 2 multiple so the number of
unrolled loops and the number of remaining limbs can be calculated with
a shift and mask. But other multiples can be used too, just by
subtracting each N limbs processed from a counter and waiting for less
than N remaining (or offsetting the counter by N so it goes negative
when there's less than N remaining).
The limbs not a multiple of the unrolling can be handled in various
ways, for example
* A simple loop at the end (or the start) to process the excess.
Care will be wanted that it isn't too much slower than the
unrolled part.
* A set of binary tests, for example after an 8-limb unrolling, test
for 4 more limbs to process, then a further 2 more or not, and
finally 1 more or not. This will probably take more code space
than a simple loop.
* A `switch' statement, providing separate code for each possible
excess, for example an 8-limb unrolling would have separate code
for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
take a lot of code, but may be the best way to optimize all cases
in combination with a deep pipelined loop.
* A computed jump into the middle of the loop, thus making the first
iteration handle the excess. This should make times smoothly
increase with size, which is attractive, but setups for the jump
and adjustments for pointers can be tricky and could become quite
difficult in combination with deep pipelining.
One way to write the setups and finishups for a pipelined unrolled
loop is simply to duplicate the loop at the start and the end, then
delete instructions at the start which have no valid antecedents, and
delete instructions at the end whose results are unwanted. Sizes not a
multiple of the unrolling can then be handled as desired.
Internals
*********
*This chapter is provided only for informational purposes and the
various internals described here may change in future GMP releases.
Applications expecting to be compatible with future releases should use
only the documented interfaces described in previous chapters.*
Integer Internals
=================
`mpz_t' variables represent integers using sign and magnitude, in
space dynamically allocated and reallocated. The fields are as follows.
`_mp_size'
The number of limbs, or the negative of that when representing a
negative integer. Zero is represented by `_mp_size' set to zero,
in which case the `_mp_d' data is unused.
`_mp_d'
A pointer to an array of limbs which is the magnitude. These are
stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
most significant. Whenever `_mp_size' is non-zero, the most
significant limb is non-zero.
Currently there's always at least one limb allocated, so for
instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
can fetch `_mp_d[0]' unconditionally (though its value is then
only wanted if `_mp_size' is non-zero).
`_mp_alloc'
`_mp_alloc' is the number of limbs currently allocated at `_mp_d',
and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine
is about to (or might be about to) increase `_mp_size', it checks
`_mp_alloc' to see whether there's enough space, and reallocates
if not. `MPZ_REALLOC' is generally used for this.
The various bitwise logical functions like `mpz_and' behave as if
negative values were twos complement. But sign and magnitude is always
used internally, and necessary adjustments are made during the
calculations. Sometimes this isn't pretty, but sign and magnitude are
best for other routines.
Some internal temporary variables are setup with `MPZ_TMP_INIT' and
these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
memory allocation functions. Care is taken to ensure that these are
big enough that no reallocation is necessary (since it would have
unpredictable consequences).
Rational Internals
==================
`mpq_t' variables represent rationals using an `mpz_t' numerator and
denominator (*note Integer Internals::).
The canonical form adopted is denominator positive (and non-zero),
no common factors between numerator and denominator, and zero uniquely
represented as 0/1.
It's believed that casting out common factors at each stage of a
calculation is best in general. A GCD is an O(N^2) operation so it's
better to do a few small ones immediately than to delay and have to do
a big one later. Knowing the numerator and denominator have no common
factors can be used for example in `mpq_mul' to make only two cross
GCDs necessary, not four.
This general approach to common factors is badly sub-optimal in the
presence of simple factorizations or little prospect for cancellation,
but GMP has no way to know when this will occur. As per *Note
Efficiency::, that's left to applications. The `mpq_t' framework might
still suit, with `mpq_numref' and `mpq_denref' for direct access to the
numerator and denominator, or of course `mpz_t' variables can be used
directly.
Float Internals
===============
Efficient calculation is the primary aim of GMP floats and the use
of whole limbs and simple rounding facilitates this.
`mpf_t' floats have a variable precision mantissa and a single
machine word signed exponent. The mantissa is represented using sign
and magnitude.
most least
significant significant
limb limb
_mp_d
|---- _mp_exp ---> |
_____ _____ _____ _____ _____
|_____|_____|_____|_____|_____|
. <------------ radix point
<-------- _mp_size --------->
The fields are as follows.
`_mp_size'
The number of limbs currently in use, or the negative of that when
representing a negative value. Zero is represented by `_mp_size'
and `_mp_exp' both set to zero, and in that case the `_mp_d' data
is unused. (In the future `_mp_exp' might be undefined when
representing zero.)
`_mp_prec'
The precision of the mantissa, in limbs. In any calculation the
aim is to produce `_mp_prec' limbs of result (the most significant
being non-zero).
`_mp_d'
A pointer to the array of limbs which is the absolute value of the
mantissa. These are stored "little endian" as per the `mpn'
functions, so `_mp_d[0]' is the least significant limb and
`_mp_d[ABS(_mp_size)-1]' the most significant.
The most significant limb is always non-zero, but there are no
other restrictions on its value, in particular the highest 1 bit
can be anywhere within the limb.
`_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
for convenience (see below). There are no reallocations during a
calculation, only in a change of precision with `mpf_set_prec'.
`_mp_exp'
The exponent, in limbs, determining the location of the implied
radix point. Zero means the radix point is just above the most
significant limb. Positive values mean a radix point offset
towards the lower limbs and hence a value >= 1, as for example in
the diagram above. Negative exponents mean a radix point further
above the highest limb.
Naturally the exponent can be any value, it doesn't have to fall
within the limbs as the diagram shows, it can be a long way above
or a long way below. Limbs other than those included in the
`{_mp_d,_mp_size}' data are treated as zero.
The following various points should be noted.
Low Zeros
The least significant limbs `_mp_d[0]' etc can be zero, though
such low zeros can always be ignored. Routines likely to produce
low zeros check and avoid them to save time in subsequent
calculations, but for most routines they're quite unlikely and
aren't checked.
Mantissa Size Range
The `_mp_size' count of limbs in use can be less than `_mp_prec' if
the value can be represented in less. This means low precision
values or small integers stored in a high precision `mpf_t' can
still be operated on efficiently.
`_mp_size' can also be greater than `_mp_prec'. Firstly a value is
allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
`_mp_size' unchanged and so the size can be arbitrarily bigger than
`_mp_prec'.
Rounding
All rounding is done on limb boundaries. Calculating `_mp_prec'
limbs with the high non-zero will ensure the application requested
minimum precision is obtained.
The use of simple "trunc" rounding towards zero is efficient,
since there's no need to examine extra limbs and increment or
decrement.
Bit Shifts
Since the exponent is in limbs, there are no bit shifts in basic
operations like `mpf_add' and `mpf_mul'. When differing exponents
are encountered all that's needed is to adjust pointers to line up
the relevant limbs.
Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
shifts, but the choice is between an exponent in limbs which
requires shifts there, or one in bits which requires them almost
everywhere else.
Use of `_mp_prec+1' Limbs
The extra limb on `_mp_d' (`_mp_prec+1' rather than just
`_mp_prec') helps when an `mpf' routine might get a carry from its
operation. `mpf_add' for instance will do an `mpn_add' of
`_mp_prec' limbs. If there's no carry then that's the result, but
if there is a carry then it's stored in the extra limb of space and
`_mp_size' becomes `_mp_prec+1'.
Whenever `_mp_prec+1' limbs are held in a variable, the low limb
is not needed for the intended precision, only the `_mp_prec' high
limbs. But zeroing it out or moving the rest down is unnecessary.
Subsequent routines reading the value will simply take the high
limbs they need, and this will be `_mp_prec' if their target has
that same precision. This is no more than a pointer adjustment,
and must be checked anyway since the destination precision can be
different from the sources.
Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
if available. This ensures that a variable which has `_mp_size'
equal to `_mp_prec+1' will get its full exact value copied.
Strictly speaking this is unnecessary since only `_mp_prec' limbs
are needed for the application's requested precision, but it's
considered that an `mpf_set' from one variable into another of the
same precision ought to produce an exact copy.
Application Precisions
`__GMPF_BITS_TO_PREC' converts an application requested precision
to an `_mp_prec'. The value in bits is rounded up to a whole limb
then an extra limb is added since the most significant limb of
`_mp_d' is only non-zero and therefore might contain only one bit.
`__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
extra limb from `_mp_prec' before converting to bits. The net
effect of reading back with `mpf_get_prec' is simply the precision
rounded up to a multiple of `mp_bits_per_limb'.
Note that the extra limb added here for the high only being
non-zero is in addition to the extra limb allocated to `_mp_d'.
For example with a 32-bit limb, an application request for 250
bits will be rounded up to 8 limbs, then an extra added for the
high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then
gets 10 limbs allocated. Reading back with `mpf_get_prec' will
take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
bits.
Strictly speaking, the fact the high limb has at least one bit
means that a float with, say, 3 limbs of 32-bits each will be
holding at least 65 bits, but for the purposes of `mpf_t' it's
considered simply to be 64 bits, a nice multiple of the limb size.
Raw Output Internals
====================
`mpz_out_raw' uses the following format.
+------+------------------------+
| size | data bytes |
+------+------------------------+
The size is 4 bytes written most significant byte first, being the
number of subsequent data bytes, or the twos complement negative of
that when a negative integer is represented. The data bytes are the
absolute value of the integer, written most significant byte first.
The most significant data byte is always non-zero, so the output is
the same on all systems, irrespective of limb size.
In GMP 1, leading zero bytes were written to pad the data bytes to a
multiple of the limb size. `mpz_inp_raw' will still accept this, for
compatibility.
The use of "big endian" for both the size and data fields is
deliberate, it makes the data easy to read in a hex dump of a file.
Unfortunately it also means that the limb data must be reversed when
reading or writing, so neither a big endian nor little endian system
can just read and write `_mp_d'.
C++ Interface Internals
=======================
A system of expression templates is used to ensure something like
`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' and
`mpfr_class' the scheme also ensures the precision of the final
destination is used for any temporaries within a statement like
`f=w*x+y*z'. These are important features which a naive implementation
cannot provide.
A simplified description of the scheme follows. The true scheme is
complicated by the fact that expressions have different return types.
For detailed information, refer to the source code.
To perform an operation, say, addition, we first define a "function
object" evaluating it,
struct __gmp_binary_plus
{
static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
};
And an "additive expression" object,
__gmp_expr<__gmp_binary_expr >
operator+(const mpf_class &f, const mpf_class &g)
{
return __gmp_expr
<__gmp_binary_expr >(f, g);
}
The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
to encapsulate any possible kind of expression into a single template
type. In fact even `mpf_class' etc are `typedef' specializations of
`__gmp_expr'.
Next we define assignment of `__gmp_expr' to `mpf_class'.
template
mpf_class & mpf_class::operator=(const __gmp_expr &expr)
{
expr.eval(this->get_mpf_t(), this->precision());
return *this;
}
template
void __gmp_expr<__gmp_binary_expr >::eval
(mpf_t f, unsigned long int precision)
{
Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
}
where `expr.val1' and `expr.val2' are references to the expression's
operands (here `expr' is the `__gmp_binary_expr' stored within the
`__gmp_expr').
This way, the expression is actually evaluated only at the time of
assignment, when the required precision (that of `f') is known.
Furthermore the target `mpf_t' is now available, thus we can call
`mpf_add' directly with `f' as the output argument.
Compound expressions are handled by defining operators taking
subexpressions as their arguments, like this:
template
__gmp_expr
<__gmp_binary_expr<__gmp_expr, __gmp_expr__, __gmp_binary_plus> >
operator+(const __gmp_expr &expr1, const __gmp_expr____ &expr2)
{
return __gmp_expr
<__gmp_binary_expr<__gmp_expr, __gmp_expr____, __gmp_binary_plus> >
(expr1, expr2);
}
And the corresponding specializations of `__gmp_expr::eval':
template
void __gmp_expr
<__gmp_binary_expr<__gmp_expr, __gmp_expr____, Op> >::eval
(mpf_t f, unsigned long int precision)
{
// declare two temporaries
mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
}
The expression is thus recursively evaluated to any level of
complexity and all subexpressions are evaluated to the precision of `f'.
Contributors
************
Torbjorn Granlund wrote the original GMP library and is still
developing and maintaining it. Several other individuals and
organizations have contributed to GMP in various ways. Here is a list
in chronological order:
Gunnar Sjoedin and Hans Riesel helped with mathematical problems in
early versions of the library.
Richard Stallman contributed to the interface design and revised the
first version of this manual.
Brian Beuning and Doug Lea helped with testing of early versions of
the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
`mpz_probab_prime_p'.
Paul Zimmermann of Inria sparked the development of GMP 2, with his
comparisons between bignum packages.
Ken Weber (Kent State University, Universidade Federal do Rio Grande
do Sul) contributed `mpz_gcd', `mpz_divexact', `mpn_gcd', and
`mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2.
Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
configure. He has also made valuable suggestions and tested numerous
intermediary releases.
Joachim Hollman was involved in the design of the `mpf' interface,
and in the `mpz' design revisions for version 2.
Bennet Yee contributed the initial versions of `mpz_jacobi' and
`mpz_legendre'.
Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
`mpn/m68k/rshift.S' (now in `.asm' form).
The development of floating point functions of GNU MP 2, were
supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
project POSSO (POlynomial System SOlving).
GNU MP 2 was finished and released by SWOX AB, SWEDEN, in
cooperation with the IDA Center for Computing Sciences, USA.
Robert Harley of Inria, France and David Seal of ARM, England,
suggested clever improvements for population count.
Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
multiplication functions for GMP 3. He also contributed the ARM
assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm
University provided significant inspiration during several phases of
the GMP development. His mathematical expertise helped improve several
algorithms.
Paul Zimmermann wrote the Divide and Conquer division code, the REDC
code, the REDC-based mpz_powm code, the FFT multiply code, and the
Karatsuba square root. The ECMNET project Paul is organizing was a
driving force behind many of the optimizations in GMP 3.
Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.
Kent Boortz made the Macintosh port.
Kevin Ryde worked on a number of things: optimized x86 code, m4 asm
macros, parameter tuning, speed measuring, the configure system,
function inlining, divisibility tests, bit scanning, Jacobi symbols,
Fibonacci and Lucas number functions, printf and scanf functions, perl
interface, demo expression parser, the algorithms chapter in the
manual, `gmpasm-mode.el', and various miscellaneous improvements
elsewhere.
Steve Root helped write the optimized alpha 21264 assembly code.
Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
`istream' input routines.
GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin
Ryde. Torbjorn's work was partially funded by the IDA Center for
Computing Sciences, USA.
(This list is chronological, not ordered after significance. If you
have contributed to GMP but are not listed above, please tell
about the omission!)
Thanks goes to Hans Thorsen for donating an SGI system for the GMP
test system environment.
References
**********
Books
=====
* Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
in Analytic Number Theory and Computational Complexity", Wiley,
John & Sons, 1998.
* Henri Cohen, "A Course in Computational Algebraic Number Theory",
Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
`http://www.math.u-bordeaux.fr/~cohen'
* Donald E. Knuth, "The Art of Computer Programming", volume 2,
"Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
`http://www-cs-faculty.stanford.edu/~knuth/taocp.html'
* John D. Lipson, "Elements of Algebra and Algebraic Computing", The
Benjamin Cummings Publishing Company Inc, 1981.
* Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
"Handbook of Applied Cryptography",
`http://www.cacr.math.uwaterloo.ca/hac/'
* Richard M. Stallman, "Using and Porting GCC", Free Software
Foundation, 1999, available online
`http://www.gnu.org/software/gcc/onlinedocs/', and in the GCC
package `ftp://ftp.gnu.org/gnu/gcc/'
Papers
======
* Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
Max-Planck-Institut fuer Informatik Research Report
MPI-I-98-1-022,
`http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'
* Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant
Integers using Multiplication", in Proceedings of the SIGPLAN
PLDI'94 Conference, June 1994. Also available
`ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz).
* Peter L. Montgomery, "Modular Multiplication Without Trial
Division", in Mathematics of Computation, volume 44, number 170,
April 1985.
* Tudor Jebelean, "An algorithm for exact division", Journal of
Symbolic Computation, volume 15, 1993, pp. 169-180. Research
report version available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
* Tudor Jebelean, "Exact Division with Karatsuba Complexity -
Extended Abstract", RISC-Linz technical report 96-31,
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'
* Tudor Jebelean, "Practical Integer Division with Karatsuba
Complexity", ISSAC 97, pp. 339-341. Technical report available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'
* Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
ISSAC 93, pp. 111-116. Technical report version available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'
* Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
Finding the GCD of Long Integers", Journal of Symbolic
Computation, volume 19, 1995, pp. 145-157. Technical report
version also available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'
* Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
Division", Journal of Symbolic Computation, volume 21, 1996, pp.
441-455. Early technical report version also available
`ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'
* R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
Proceedings of the 13th Annual IEEE Symposium on Switching and
Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast
Modular Transforms", Journal of Computer and System Sciences,
volume 8, number 3, June 1974, pp. 366-386.
* Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation
grosser Zahlen", Computing 7, 1971, pp. 281-292.
* Kenneth Weber, "The accelerated integer GCD algorithm", ACM
Transactions on Mathematical Software, volume 21, number 1, March
1995, pp. 111-122.
* Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
3805, November 1999, `http://www.inria.fr/RRRT/RR-3805.html'
* Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
Implementations",
`http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
* Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
Reprinted as "More on Multiplying and Squaring Large Integers",
IEEE Transactions on Computers, volume 43, number 8, August 1994,
pp. 899-908.
GNU Free Documentation License
******************************
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ADDENDUM: How to use this License for your documents
====================================================
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Concept Index
*************
ABI:
See ``ABI and ISA''.
About this manual:
See ``Introduction to GNU MP''.
Algorithms:
See ``Algorithms''.
alloca:
See ``Build Options''.
Allocation of memory:
See ``Custom Allocation''.
Anonymous FTP of latest version:
See ``Introduction to GNU MP''.
Application Binary Interface:
See ``ABI and ISA''.
Arithmetic functions <1>:
See ``Arithmetic Functions''.
Arithmetic functions <2>:
See ``Arithmetic Functions''.
Arithmetic functions:
See ``Arithmetic Functions''.
Assignment functions <1>:
See ``Assignment Functions''.
Assignment functions:
See ``Assignment Functions''.
Autoconf detections:
See ``Autoconf''.
Basics:
See ``GMP Basics''.
Berkeley MP compatible functions:
See ``Berkeley MP Compatible Functions''.
Binomial coefficient functions:
See ``Number Theoretic Functions''.
Bit manipulation functions:
See ``Logical and Bit Manipulation Functions''.
Bit shift left:
See ``Arithmetic Functions''.
Bit shift right:
See ``Division Functions''.
Bits per limb:
See ``Useful Macros and Constants''.
BSD MP compatible functions:
See ``Berkeley MP Compatible Functions''.
Bug reporting:
See ``Reporting Bugs''.
Build notes for binary packaging:
See ``Notes for Package Builds''.
Build notes for particular systems:
See ``Notes for Particular Systems''.
Build options:
See ``Build Options''.
Build problems known:
See ``Known Build Problems''.
Building GMP:
See ``Installing GMP''.
C++ Interface:
See ``C++ Class Interface''.
C++ istream input:
See ``C++ Formatted Input''.
C++ ostream output:
See ``C++ Formatted Output''.
Comparison functions <1>:
See ``Comparison Functions''.
Comparison functions <2>:
See ``Comparison Functions''.
Comparison functions:
See ``Comparison Functions''.
Compatibility with older versions:
See ``Compatibility with older versions''.
Conditions for copying GNU MP:
See ``GNU MP Copying Conditions''.
Configuring GMP:
See ``Installing GMP''.
Constants:
See ``Useful Macros and Constants''.
Contributors:
See ``Contributors''.
Conventions for parameters:
See ``Parameter Conventions''.
Conventions for variables:
See ``Variable Conventions''.
Conversion functions <1>:
See ``Conversion Functions''.
Conversion functions <2>:
See ``Conversion Functions''.
Conversion functions:
See ``Conversion Functions''.
Copying conditions:
See ``GNU MP Copying Conditions''.
CPUs supported:
See ``Introduction to GNU MP''.
Custom allocation:
See ``Custom Allocation''.
Debugging:
See ``Debugging''.
Demonstration programs:
See ``Build Options''.
DESTDIR:
See ``Known Build Problems''.
Division algorithms:
See ``Division Algorithms''.
Division functions <1>:
See ``Arithmetic Functions''.
Division functions <2>:
See ``Arithmetic Functions''.
Division functions:
See ``Division Functions''.
Efficiency:
See ``Efficiency''.
Exact division functions:
See ``Division Functions''.
Example programs:
See ``Build Options''.
Exponentiation functions <1>:
See ``Arithmetic Functions''.
Exponentiation functions:
See ``Exponentiation Functions''.
Export:
See ``Integer Import and Export''.
Extended GCD:
See ``Number Theoretic Functions''.
Factorial functions:
See ``Number Theoretic Functions''.
FDL, GNU Free Documentation License:
See ``GNU Free Documentation License''.
Fibonacci sequence functions:
See ``Number Theoretic Functions''.
Float arithmetic functions:
See ``Arithmetic Functions''.
Float assignment functions:
See ``Assignment Functions''.
Float comparison functions:
See ``Comparison Functions''.
Float conversion functions:
See ``Conversion Functions''.
Float functions:
See ``Floating-point Functions''.
Float init and assign functions:
See ``Combined Initialization and Assignment Functions''.
Float initialization functions:
See ``Initialization Functions''.
Float input and output functions:
See ``Input and Output Functions''.
Float miscellaneous functions:
See ``Miscellaneous Functions''.
Float sign tests:
See ``Comparison Functions''.
Floating-point functions:
See ``Floating-point Functions''.
Floating-point number:
See ``Nomenclature and Types''.
Formatted input:
See ``Formatted Input''.
Formatted output:
See ``Formatted Output''.
FTP of latest version:
See ``Introduction to GNU MP''.
Function classes:
See ``Function Classes''.
GMP version number:
See ``Useful Macros and Constants''.
gmp.h:
See ``Headers and Libraries''.
gmpxx.h:
See ``C++ Interface General''.
GNU Free Documentation License:
See ``GNU Free Documentation License''.
Greatest common divisor algorithms:
See ``Greatest Common Divisor''.
Greatest common divisor functions:
See ``Number Theoretic Functions''.
Headers:
See ``Headers and Libraries''.
Home page:
See ``Introduction to GNU MP''.
I/O functions <1>:
See ``Input and Output Functions''.
I/O functions <2>:
See ``Input and Output Functions''.
I/O functions:
See ``Input and Output Functions''.
Import:
See ``Integer Import and Export''.
Initialization and assignment functions <1>:
See ``Combined Initialization and Assignment Functions''.
Initialization and assignment functions <2>:
See ``Initialization and Assignment Functions''.
Initialization and assignment functions:
See ``Combined Initialization and Assignment Functions''.
Initialization functions <1>:
See ``Initialization Functions''.
Initialization functions:
See ``Initialization Functions''.
Input functions <1>:
See ``Input and Output Functions''.
Input functions <2>:
See ``Input and Output Functions''.
Input functions:
See ``Input and Output Functions''.
Installing GMP:
See ``Installing GMP''.
Instruction Set Architecture:
See ``ABI and ISA''.
Integer:
See ``Nomenclature and Types''.
Integer arithmetic functions:
See ``Arithmetic Functions''.
Integer assignment functions:
See ``Assignment Functions''.
Integer bit manipulation functions:
See ``Logical and Bit Manipulation Functions''.
Integer comparison functions:
See ``Comparison Functions''.
Integer conversion functions:
See ``Conversion Functions''.
Integer division functions:
See ``Division Functions''.
Integer exponentiation functions:
See ``Exponentiation Functions''.
Integer export:
See ``Integer Import and Export''.
Integer functions:
See ``Integer Functions''.
Integer import:
See ``Integer Import and Export''.
Integer init and assign:
See ``Combined Initialization and Assignment Functions''.
Integer initialization functions:
See ``Initialization Functions''.
Integer input and output functions:
See ``Input and Output Functions''.
Integer miscellaneous functions:
See ``Miscellaneous Functions''.
Integer random number functions:
See ``Random Number Functions''.
Integer root functions:
See ``Root Extraction Functions''.
Integer sign tests:
See ``Comparison Functions''.
Introduction:
See ``Introduction to GNU MP''.
ISA:
See ``ABI and ISA''.
istream input:
See ``C++ Formatted Input''.
Jacobi symbol functions:
See ``Number Theoretic Functions''.
Kronecker symbol functions:
See ``Number Theoretic Functions''.
Latest version of GMP:
See ``Introduction to GNU MP''.
Least common multiple functions:
See ``Number Theoretic Functions''.
Libraries:
See ``Headers and Libraries''.
Libtool versioning:
See ``Notes for Package Builds''.
License conditions:
See ``GNU MP Copying Conditions''.
Limb:
See ``Nomenclature and Types''.
Limb size:
See ``Useful Macros and Constants''.
Linking:
See ``Headers and Libraries''.
Logical functions:
See ``Logical and Bit Manipulation Functions''.
Low-level functions:
See ``Low-level Functions''.
Lucas number functions:
See ``Number Theoretic Functions''.
Mailing list:
See ``Introduction to GNU MP''.
Memory allocation:
See ``Custom Allocation''.
Memory Management:
See ``Memory Management''.
Miscellaneous float functions:
See ``Miscellaneous Functions''.
Miscellaneous integer functions:
See ``Miscellaneous Functions''.
Modular inverse functions:
See ``Number Theoretic Functions''.
mp.h:
See ``Berkeley MP Compatible Functions''.
MPFR:
See ``Build Options''.
mpfrxx.h:
See ``C++ Interface MPFR''.
Multi-threading:
See ``Reentrancy''.
Multiplication algorithms:
See ``Multiplication''.
Nails:
See ``Low-level Functions''.
Nomenclature:
See ``Nomenclature and Types''.
Number theoretic functions:
See ``Number Theoretic Functions''.
Numerator and denominator:
See ``Applying Integer Functions to Rationals''.
ostream output:
See ``C++ Formatted Output''.
Output functions <1>:
See ``Input and Output Functions''.
Output functions <2>:
See ``Input and Output Functions''.
Output functions:
See ``Input and Output Functions''.
Packaged builds:
See ``Notes for Package Builds''.
Parameter conventions:
See ``Parameter Conventions''.
Particular systems:
See ``Notes for Particular Systems''.
Powering algorithms:
See ``Powering Algorithms''.
Powering functions <1>:
See ``Arithmetic Functions''.
Powering functions:
See ``Exponentiation Functions''.
Precision of floats:
See ``Floating-point Functions''.
Prime testing functions:
See ``Number Theoretic Functions''.
printf formatted output:
See ``Formatted Output''.
Profiling:
See ``Profiling''.
Radix conversion algorithms:
See ``Radix Conversion''.
Random number functions <1>:
See ``Random Number Functions''.
Random number functions:
See ``Random Number Functions''.
Random number seeding:
See ``Random State Seeding''.
Random number state:
See ``Random State Initialization''.
Rational arithmetic functions:
See ``Arithmetic Functions''.
Rational comparison functions:
See ``Comparison Functions''.
Rational conversion functions:
See ``Conversion Functions''.
Rational init and assign:
See ``Initialization and Assignment Functions''.
Rational input and output functions:
See ``Input and Output Functions''.
Rational number:
See ``Nomenclature and Types''.
Rational number functions:
See ``Rational Number Functions''.
Rational numerator and denominator:
See ``Applying Integer Functions to Rationals''.
Rational sign tests:
See ``Comparison Functions''.
Reentrancy:
See ``Reentrancy''.
References:
See ``References''.
Reporting bugs:
See ``Reporting Bugs''.
Root extraction algorithms:
See ``Root Extraction Algorithms''.
Root extraction functions <1>:
See ``Arithmetic Functions''.
Root extraction functions:
See ``Root Extraction Functions''.
scanf formatted input:
See ``Formatted Input''.
Shared library versioning:
See ``Notes for Package Builds''.
Sign tests <1>:
See ``Comparison Functions''.
Sign tests <2>:
See ``Comparison Functions''.
Sign tests:
See ``Comparison Functions''.
Stack overflow segfaults:
See ``Build Options''.
Stripped libraries:
See ``Known Build Problems''.
Systems:
See ``Notes for Particular Systems''.
Thread safety:
See ``Reentrancy''.
Types:
See ``Nomenclature and Types''.
Upward compatibility:
See ``Compatibility with older versions''.
Useful macros and constants:
See ``Useful Macros and Constants''.
User-defined precision:
See ``Floating-point Functions''.
Variable conventions:
See ``Variable Conventions''.
Version number:
See ``Useful Macros and Constants''.
Web page:
See ``Introduction to GNU MP''.
Function and Type Index
***********************
*mpz_export:
See ``Integer Import and Export''.
__GNU_MP_VERSION:
See ``Useful Macros and Constants''.
__GNU_MP_VERSION_MINOR:
See ``Useful Macros and Constants''.
__GNU_MP_VERSION_PATCHLEVEL:
See ``Useful Macros and Constants''.
_mpz_realloc:
See ``Initialization Functions''.
abs <1>:
See ``C++ Interface Floats''.
abs <2>:
See ``C++ Interface Rationals''.
abs:
See ``C++ Interface Integers''.
allocate_function:
See ``Custom Allocation''.
ceil:
See ``C++ Interface Floats''.
cmp <1>:
See ``C++ Interface Floats''.
cmp <2>:
See ``C++ Interface Rationals''.
cmp:
See ``C++ Interface Integers''.
deallocate_function:
See ``Custom Allocation''.
floor:
See ``C++ Interface Floats''.
gcd:
See ``Berkeley MP Compatible Functions''.
gmp_asprintf:
See ``Functions''.
gmp_fprintf:
See ``Functions''.
gmp_fscanf:
See ``Formatted Input Functions''.
GMP_LIMB_BITS:
See ``Low-level Functions''.
GMP_NAIL_BITS:
See ``Low-level Functions''.
GMP_NAIL_MASK:
See ``Low-level Functions''.
GMP_NUMB_BITS:
See ``Low-level Functions''.
GMP_NUMB_MASK:
See ``Low-level Functions''.
GMP_NUMB_MAX:
See ``Low-level Functions''.
gmp_obstack_printf:
See ``Functions''.
gmp_obstack_vprintf:
See ``Functions''.
gmp_printf:
See ``Functions''.
gmp_randclass:
See ``C++ Interface Random Numbers''.
gmp_randclass::get_f:
See ``C++ Interface Random Numbers''.
gmp_randclass::get_z_bits:
See ``C++ Interface Random Numbers''.
gmp_randclass::get_z_range:
See ``C++ Interface Random Numbers''.
gmp_randclass::gmp_randclass:
See ``C++ Interface Random Numbers''.
gmp_randclass::seed:
See ``C++ Interface Random Numbers''.
gmp_randclear:
See ``Random State Initialization''.
gmp_randinit:
See ``Random State Initialization''.
gmp_randinit_default:
See ``Random State Initialization''.
gmp_randinit_lc_2exp:
See ``Random State Initialization''.
gmp_randinit_lc_2exp_size:
See ``Random State Initialization''.
gmp_randseed:
See ``Random State Seeding''.
gmp_randseed_ui:
See ``Random State Seeding''.
gmp_scanf:
See ``Formatted Input Functions''.
gmp_snprintf:
See ``Functions''.
gmp_sprintf:
See ``Functions''.
gmp_sscanf:
See ``Formatted Input Functions''.
gmp_vasprintf:
See ``Functions''.
gmp_version:
See ``Useful Macros and Constants''.
gmp_vfprintf:
See ``Functions''.
gmp_vfscanf:
See ``Formatted Input Functions''.
gmp_vprintf:
See ``Functions''.
gmp_vscanf:
See ``Formatted Input Functions''.
gmp_vsnprintf:
See ``Functions''.
gmp_vsprintf:
See ``Functions''.
gmp_vsscanf:
See ``Formatted Input Functions''.
hypot:
See ``C++ Interface Floats''.
itom:
See ``Berkeley MP Compatible Functions''.
madd:
See ``Berkeley MP Compatible Functions''.
mcmp:
See ``Berkeley MP Compatible Functions''.
mdiv:
See ``Berkeley MP Compatible Functions''.
mfree:
See ``Berkeley MP Compatible Functions''.
min:
See ``Berkeley MP Compatible Functions''.
mout:
See ``Berkeley MP Compatible Functions''.
move:
See ``Berkeley MP Compatible Functions''.
mp_bits_per_limb:
See ``Useful Macros and Constants''.
mp_limb_t:
See ``Nomenclature and Types''.
mp_set_memory_functions:
See ``Custom Allocation''.
mpf_abs:
See ``Arithmetic Functions''.
mpf_add:
See ``Arithmetic Functions''.
mpf_add_ui:
See ``Arithmetic Functions''.
mpf_ceil:
See ``Miscellaneous Functions''.
mpf_class:
See ``C++ Interface General''.
mpf_class::fits_sint_p:
See ``C++ Interface Floats''.
mpf_class::fits_slong_p:
See ``C++ Interface Floats''.
mpf_class::fits_sshort_p:
See ``C++ Interface Floats''.
mpf_class::fits_uint_p:
See ``C++ Interface Floats''.
mpf_class::fits_ulong_p:
See ``C++ Interface Floats''.
mpf_class::fits_ushort_p:
See ``C++ Interface Floats''.
mpf_class::get_d:
See ``C++ Interface Floats''.
mpf_class::get_mpf_t:
See ``C++ Interface General''.
mpf_class::get_prec:
See ``C++ Interface Floats''.
mpf_class::get_si:
See ``C++ Interface Floats''.
mpf_class::get_ui:
See ``C++ Interface Floats''.
mpf_class::mpf_class:
See ``C++ Interface Floats''.
mpf_class::set_prec:
See ``C++ Interface Floats''.
mpf_class::set_prec_raw:
See ``C++ Interface Floats''.
mpf_clear:
See ``Initialization Functions''.
mpf_cmp:
See ``Comparison Functions''.
mpf_cmp_d:
See ``Comparison Functions''.
mpf_cmp_si:
See ``Comparison Functions''.
mpf_cmp_ui:
See ``Comparison Functions''.
mpf_div:
See ``Arithmetic Functions''.
mpf_div_2exp:
See ``Arithmetic Functions''.
mpf_div_ui:
See ``Arithmetic Functions''.
mpf_eq:
See ``Comparison Functions''.
mpf_fits_sint_p:
See ``Miscellaneous Functions''.
mpf_fits_slong_p:
See ``Miscellaneous Functions''.
mpf_fits_sshort_p:
See ``Miscellaneous Functions''.
mpf_fits_uint_p:
See ``Miscellaneous Functions''.
mpf_fits_ulong_p:
See ``Miscellaneous Functions''.
mpf_fits_ushort_p:
See ``Miscellaneous Functions''.
mpf_floor:
See ``Miscellaneous Functions''.
mpf_get_d:
See ``Conversion Functions''.
mpf_get_d_2exp:
See ``Conversion Functions''.
mpf_get_default_prec:
See ``Initialization Functions''.
mpf_get_prec:
See ``Initialization Functions''.
mpf_get_si:
See ``Conversion Functions''.
mpf_get_str:
See ``Conversion Functions''.
mpf_get_ui:
See ``Conversion Functions''.
mpf_init:
See ``Initialization Functions''.
mpf_init2:
See ``Initialization Functions''.
mpf_init_set:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_d:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_si:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_str:
See ``Combined Initialization and Assignment Functions''.
mpf_init_set_ui:
See ``Combined Initialization and Assignment Functions''.
mpf_inp_str:
See ``Input and Output Functions''.
mpf_integer_p:
See ``Miscellaneous Functions''.
mpf_mul:
See ``Arithmetic Functions''.
mpf_mul_2exp:
See ``Arithmetic Functions''.
mpf_mul_ui:
See ``Arithmetic Functions''.
mpf_neg:
See ``Arithmetic Functions''.
mpf_out_str:
See ``Input and Output Functions''.
mpf_pow_ui:
See ``Arithmetic Functions''.
mpf_random2:
See ``Miscellaneous Functions''.
mpf_reldiff:
See ``Comparison Functions''.
mpf_set:
See ``Assignment Functions''.
mpf_set_d:
See ``Assignment Functions''.
mpf_set_default_prec:
See ``Initialization Functions''.
mpf_set_prec:
See ``Initialization Functions''.
mpf_set_prec_raw:
See ``Initialization Functions''.
mpf_set_q:
See ``Assignment Functions''.
mpf_set_si:
See ``Assignment Functions''.
mpf_set_str:
See ``Assignment Functions''.
mpf_set_ui:
See ``Assignment Functions''.
mpf_set_z:
See ``Assignment Functions''.
mpf_sgn:
See ``Comparison Functions''.
mpf_sqrt:
See ``Arithmetic Functions''.
mpf_sqrt_ui:
See ``Arithmetic Functions''.
mpf_sub:
See ``Arithmetic Functions''.
mpf_sub_ui:
See ``Arithmetic Functions''.
mpf_swap:
See ``Assignment Functions''.
mpf_t:
See ``Nomenclature and Types''.
mpf_trunc:
See ``Miscellaneous Functions''.
mpf_ui_div:
See ``Arithmetic Functions''.
mpf_ui_sub:
See ``Arithmetic Functions''.
mpf_urandomb:
See ``Miscellaneous Functions''.
mpfr_class:
See ``C++ Interface MPFR''.
mpn_add:
See ``Low-level Functions''.
mpn_add_1:
See ``Low-level Functions''.
mpn_add_n:
See ``Low-level Functions''.
mpn_addmul_1:
See ``Low-level Functions''.
mpn_bdivmod:
See ``Low-level Functions''.
mpn_cmp:
See ``Low-level Functions''.
mpn_divexact_by3:
See ``Low-level Functions''.
mpn_divexact_by3c:
See ``Low-level Functions''.
mpn_divmod:
See ``Low-level Functions''.
mpn_divmod_1:
See ``Low-level Functions''.
mpn_divrem:
See ``Low-level Functions''.
mpn_divrem_1:
See ``Low-level Functions''.
mpn_gcd:
See ``Low-level Functions''.
mpn_gcd_1:
See ``Low-level Functions''.
mpn_gcdext:
See ``Low-level Functions''.
mpn_get_str:
See ``Low-level Functions''.
mpn_hamdist:
See ``Low-level Functions''.
mpn_lshift:
See ``Low-level Functions''.
mpn_mod_1:
See ``Low-level Functions''.
mpn_mul:
See ``Low-level Functions''.
mpn_mul_1:
See ``Low-level Functions''.
mpn_mul_n:
See ``Low-level Functions''.
mpn_perfect_square_p:
See ``Low-level Functions''.
mpn_popcount:
See ``Low-level Functions''.
mpn_random:
See ``Low-level Functions''.
mpn_random2:
See ``Low-level Functions''.
mpn_rshift:
See ``Low-level Functions''.
mpn_scan0:
See ``Low-level Functions''.
mpn_scan1:
See ``Low-level Functions''.
mpn_set_str:
See ``Low-level Functions''.
mpn_sqrtrem:
See ``Low-level Functions''.
mpn_sub:
See ``Low-level Functions''.
mpn_sub_1:
See ``Low-level Functions''.
mpn_sub_n:
See ``Low-level Functions''.
mpn_submul_1:
See ``Low-level Functions''.
mpn_tdiv_qr:
See ``Low-level Functions''.
mpq_abs:
See ``Arithmetic Functions''.
mpq_add:
See ``Arithmetic Functions''.
mpq_canonicalize:
See ``Rational Number Functions''.
mpq_class:
See ``C++ Interface General''.
mpq_class::canonicalize:
See ``C++ Interface Rationals''.
mpq_class::get_d:
See ``C++ Interface Rationals''.
mpq_class::get_den:
See ``C++ Interface Rationals''.
mpq_class::get_den_mpz_t:
See ``C++ Interface Rationals''.
mpq_class::get_mpq_t:
See ``C++ Interface General''.
mpq_class::get_num:
See ``C++ Interface Rationals''.
mpq_class::get_num_mpz_t:
See ``C++ Interface Rationals''.
mpq_class::mpq_class:
See ``C++ Interface Rationals''.
mpq_clear:
See ``Initialization and Assignment Functions''.
mpq_cmp:
See ``Comparison Functions''.
mpq_cmp_si:
See ``Comparison Functions''.
mpq_cmp_ui:
See ``Comparison Functions''.
mpq_denref:
See ``Applying Integer Functions to Rationals''.
mpq_div:
See ``Arithmetic Functions''.
mpq_div_2exp:
See ``Arithmetic Functions''.
mpq_equal:
See ``Comparison Functions''.
mpq_get_d:
See ``Conversion Functions''.
mpq_get_den:
See ``Applying Integer Functions to Rationals''.
mpq_get_num:
See ``Applying Integer Functions to Rationals''.
mpq_get_str:
See ``Conversion Functions''.
mpq_init:
See ``Initialization and Assignment Functions''.
mpq_inp_str:
See ``Input and Output Functions''.
mpq_inv:
See ``Arithmetic Functions''.
mpq_mul:
See ``Arithmetic Functions''.
mpq_mul_2exp:
See ``Arithmetic Functions''.
mpq_neg:
See ``Arithmetic Functions''.
mpq_numref:
See ``Applying Integer Functions to Rationals''.
mpq_out_str:
See ``Input and Output Functions''.
mpq_set:
See ``Initialization and Assignment Functions''.
mpq_set_d:
See ``Conversion Functions''.
mpq_set_den:
See ``Applying Integer Functions to Rationals''.
mpq_set_f:
See ``Conversion Functions''.
mpq_set_num:
See ``Applying Integer Functions to Rationals''.
mpq_set_si:
See ``Initialization and Assignment Functions''.
mpq_set_str:
See ``Initialization and Assignment Functions''.
mpq_set_ui:
See ``Initialization and Assignment Functions''.
mpq_set_z:
See ``Initialization and Assignment Functions''.
mpq_sgn:
See ``Comparison Functions''.
mpq_sub:
See ``Arithmetic Functions''.
mpq_swap:
See ``Initialization and Assignment Functions''.
mpq_t:
See ``Nomenclature and Types''.
mpz_abs:
See ``Arithmetic Functions''.
mpz_add:
See ``Arithmetic Functions''.
mpz_add_ui:
See ``Arithmetic Functions''.
mpz_addmul:
See ``Arithmetic Functions''.
mpz_addmul_ui:
See ``Arithmetic Functions''.
mpz_and:
See ``Logical and Bit Manipulation Functions''.
mpz_array_init:
See ``Initialization Functions''.
mpz_bin_ui:
See ``Number Theoretic Functions''.
mpz_bin_uiui:
See ``Number Theoretic Functions''.
mpz_cdiv_q:
See ``Division Functions''.
mpz_cdiv_q_2exp:
See ``Division Functions''.
mpz_cdiv_q_ui:
See ``Division Functions''.
mpz_cdiv_qr:
See ``Division Functions''.
mpz_cdiv_qr_ui:
See ``Division Functions''.
mpz_cdiv_r:
See ``Division Functions''.
mpz_cdiv_r_2exp:
See ``Division Functions''.
mpz_cdiv_r_ui:
See ``Division Functions''.
mpz_cdiv_ui:
See ``Division Functions''.
mpz_class:
See ``C++ Interface General''.
mpz_class::fits_sint_p:
See ``C++ Interface Integers''.
mpz_class::fits_slong_p:
See ``C++ Interface Integers''.
mpz_class::fits_sshort_p:
See ``C++ Interface Integers''.
mpz_class::fits_uint_p:
See ``C++ Interface Integers''.
mpz_class::fits_ulong_p:
See ``C++ Interface Integers''.
mpz_class::fits_ushort_p:
See ``C++ Interface Integers''.
mpz_class::get_d:
See ``C++ Interface Integers''.
mpz_class::get_mpz_t:
See ``C++ Interface General''.
mpz_class::get_si:
See ``C++ Interface Integers''.
mpz_class::get_ui:
See ``C++ Interface Integers''.
mpz_class::mpz_class:
See ``C++ Interface Integers''.
mpz_clear:
See ``Initialization Functions''.
mpz_clrbit:
See ``Logical and Bit Manipulation Functions''.
mpz_cmp:
See ``Comparison Functions''.
mpz_cmp_d:
See ``Comparison Functions''.
mpz_cmp_si:
See ``Comparison Functions''.
mpz_cmp_ui:
See ``Comparison Functions''.
mpz_cmpabs:
See ``Comparison Functions''.
mpz_cmpabs_d:
See ``Comparison Functions''.
mpz_cmpabs_ui:
See ``Comparison Functions''.
mpz_com:
See ``Logical and Bit Manipulation Functions''.
mpz_congruent_2exp_p:
See ``Division Functions''.
mpz_congruent_p:
See ``Division Functions''.
mpz_congruent_ui_p:
See ``Division Functions''.
mpz_divexact:
See ``Division Functions''.
mpz_divexact_ui:
See ``Division Functions''.
mpz_divisible_2exp_p:
See ``Division Functions''.
mpz_divisible_p:
See ``Division Functions''.
mpz_divisible_ui_p:
See ``Division Functions''.
mpz_even_p:
See ``Miscellaneous Functions''.
mpz_fac_ui:
See ``Number Theoretic Functions''.
mpz_fdiv_q:
See ``Division Functions''.
mpz_fdiv_q_2exp:
See ``Division Functions''.
mpz_fdiv_q_ui:
See ``Division Functions''.
mpz_fdiv_qr:
See ``Division Functions''.
mpz_fdiv_qr_ui:
See ``Division Functions''.
mpz_fdiv_r:
See ``Division Functions''.
mpz_fdiv_r_2exp:
See ``Division Functions''.
mpz_fdiv_r_ui:
See ``Division Functions''.
mpz_fdiv_ui:
See ``Division Functions''.
mpz_fib2_ui:
See ``Number Theoretic Functions''.
mpz_fib_ui:
See ``Number Theoretic Functions''.
mpz_fits_sint_p:
See ``Miscellaneous Functions''.
mpz_fits_slong_p:
See ``Miscellaneous Functions''.
mpz_fits_sshort_p:
See ``Miscellaneous Functions''.
mpz_fits_uint_p:
See ``Miscellaneous Functions''.
mpz_fits_ulong_p:
See ``Miscellaneous Functions''.
mpz_fits_ushort_p:
See ``Miscellaneous Functions''.
mpz_gcd:
See ``Number Theoretic Functions''.
mpz_gcd_ui:
See ``Number Theoretic Functions''.
mpz_gcdext:
See ``Number Theoretic Functions''.
mpz_get_d:
See ``Conversion Functions''.
mpz_get_d_2exp:
See ``Conversion Functions''.
mpz_get_si:
See ``Conversion Functions''.
mpz_get_str:
See ``Conversion Functions''.
mpz_get_ui:
See ``Conversion Functions''.
mpz_getlimbn:
See ``Conversion Functions''.
mpz_hamdist:
See ``Logical and Bit Manipulation Functions''.
mpz_import:
See ``Integer Import and Export''.
mpz_init:
See ``Initialization Functions''.
mpz_init2:
See ``Initialization Functions''.
mpz_init_set:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_d:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_si:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_str:
See ``Combined Initialization and Assignment Functions''.
mpz_init_set_ui:
See ``Combined Initialization and Assignment Functions''.
mpz_inp_raw:
See ``Input and Output Functions''.
mpz_inp_str:
See ``Input and Output Functions''.
mpz_invert:
See ``Number Theoretic Functions''.
mpz_ior:
See ``Logical and Bit Manipulation Functions''.
mpz_jacobi:
See ``Number Theoretic Functions''.
mpz_kronecker:
See ``Number Theoretic Functions''.
mpz_kronecker_si:
See ``Number Theoretic Functions''.
mpz_kronecker_ui:
See ``Number Theoretic Functions''.
mpz_lcm:
See ``Number Theoretic Functions''.
mpz_lcm_ui:
See ``Number Theoretic Functions''.
mpz_legendre:
See ``Number Theoretic Functions''.
mpz_lucnum2_ui:
See ``Number Theoretic Functions''.
mpz_lucnum_ui:
See ``Number Theoretic Functions''.
mpz_mod:
See ``Division Functions''.
mpz_mod_ui:
See ``Division Functions''.
mpz_mul:
See ``Arithmetic Functions''.
mpz_mul_2exp:
See ``Arithmetic Functions''.
mpz_mul_si:
See ``Arithmetic Functions''.
mpz_mul_ui:
See ``Arithmetic Functions''.
mpz_neg:
See ``Arithmetic Functions''.
mpz_nextprime:
See ``Number Theoretic Functions''.
mpz_odd_p:
See ``Miscellaneous Functions''.
mpz_out_raw:
See ``Input and Output Functions''.
mpz_out_str:
See ``Input and Output Functions''.
mpz_perfect_power_p:
See ``Root Extraction Functions''.
mpz_perfect_square_p:
See ``Root Extraction Functions''.
mpz_popcount:
See ``Logical and Bit Manipulation Functions''.
mpz_pow_ui:
See ``Exponentiation Functions''.
mpz_powm:
See ``Exponentiation Functions''.
mpz_powm_ui:
See ``Exponentiation Functions''.
mpz_probab_prime_p:
See ``Number Theoretic Functions''.
mpz_random:
See ``Random Number Functions''.
mpz_random2:
See ``Random Number Functions''.
mpz_realloc2:
See ``Initialization Functions''.
mpz_remove:
See ``Number Theoretic Functions''.
mpz_root:
See ``Root Extraction Functions''.
mpz_rrandomb:
See ``Random Number Functions''.
mpz_scan0:
See ``Logical and Bit Manipulation Functions''.
mpz_scan1:
See ``Logical and Bit Manipulation Functions''.
mpz_set:
See ``Assignment Functions''.
mpz_set_d:
See ``Assignment Functions''.
mpz_set_f:
See ``Assignment Functions''.
mpz_set_q:
See ``Assignment Functions''.
mpz_set_si:
See ``Assignment Functions''.
mpz_set_str:
See ``Assignment Functions''.
mpz_set_ui:
See ``Assignment Functions''.
mpz_setbit:
See ``Logical and Bit Manipulation Functions''.
mpz_sgn:
See ``Comparison Functions''.
mpz_si_kronecker:
See ``Number Theoretic Functions''.
mpz_size:
See ``Miscellaneous Functions''.
mpz_sizeinbase:
See ``Miscellaneous Functions''.
mpz_sqrt:
See ``Root Extraction Functions''.
mpz_sqrtrem:
See ``Root Extraction Functions''.
mpz_sub:
See ``Arithmetic Functions''.
mpz_sub_ui:
See ``Arithmetic Functions''.
mpz_submul:
See ``Arithmetic Functions''.
mpz_submul_ui:
See ``Arithmetic Functions''.
mpz_swap:
See ``Assignment Functions''.
mpz_t:
See ``Nomenclature and Types''.
mpz_tdiv_q:
See ``Division Functions''.
mpz_tdiv_q_2exp:
See ``Division Functions''.
mpz_tdiv_q_ui:
See ``Division Functions''.
mpz_tdiv_qr:
See ``Division Functions''.
mpz_tdiv_qr_ui:
See ``Division Functions''.
mpz_tdiv_r:
See ``Division Functions''.
mpz_tdiv_r_2exp:
See ``Division Functions''.
mpz_tdiv_r_ui:
See ``Division Functions''.
mpz_tdiv_ui:
See ``Division Functions''.
mpz_tstbit:
See ``Logical and Bit Manipulation Functions''.
mpz_ui_kronecker:
See ``Number Theoretic Functions''.
mpz_ui_pow_ui:
See ``Exponentiation Functions''.
mpz_ui_sub:
See ``Arithmetic Functions''.
mpz_urandomb:
See ``Random Number Functions''.
mpz_urandomm:
See ``Random Number Functions''.
mpz_xor:
See ``Logical and Bit Manipulation Functions''.
msqrt:
See ``Berkeley MP Compatible Functions''.
msub:
See ``Berkeley MP Compatible Functions''.
mtox:
See ``Berkeley MP Compatible Functions''.
mult:
See ``Berkeley MP Compatible Functions''.
operator%:
See ``C++ Interface Integers''.
operator/:
See ``C++ Interface Integers''.
operator<<:
See ``C++ Formatted Output''.
operator>> <1>:
See ``C++ Interface Rationals''.
operator>>:
See ``C++ Formatted Input''.
pow:
See ``Berkeley MP Compatible Functions''.
reallocate_function:
See ``Custom Allocation''.
rpow:
See ``Berkeley MP Compatible Functions''.
sdiv:
See ``Berkeley MP Compatible Functions''.
sgn <1>:
See ``C++ Interface Floats''.
sgn <2>:
See ``C++ Interface Rationals''.
sgn:
See ``C++ Interface Integers''.
sqrt <1>:
See ``C++ Interface Floats''.
sqrt:
See ``C++ Interface Integers''.
trunc:
See ``C++ Interface Floats''.
xtom:
See ``Berkeley MP Compatible Functions''.
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